5391:. The first and most important property is that Lyndon words are a special case of Hall words. That is, the definition of a Lyndon word satisfies the definition of the Hall word. This can be readily verified by direct comparison; note that the direction of the inequality used in the definitions above is exactly the reverse of that used in the customary definition for Lyndon words. The set of Lyndon trees (which follow from the standard factorization) is a Hall set.
1876:
1454:
2493:
are obtained from the Hall set by "forgetting" the commutator brackets, but otherwise keeping the notion of total order. It turns out that this "forgetting" is harmless, as the corresponding Hall tree can be deduced from the word, and it is unique. That is, the Hall words are in one-to-one
2158:
1294:, and so can be directly compared; the weights are the string-lengths. The difference is that no mention of groups is required. These definitions all coincide with that of X. Viennot. Note that some authors reverse the order of the inequality. Note also that one of the conditions, the
4820:
It is not hard to show that both rewrites result in a new standard sequence. In general, it is most convenient to apply the rewrite to the right-most legal drop; however, it can be shown that the rewrite is confluent, and so one obtains the same results no matter what the order.
4815:
324:
provides a convenient stand-in for any other desired binary operator that might have additional properties, such as group or algebra commutators. Thus, for example, the magma juxtaposition can be mapped to the commutator of a non-commutative algebra:
4617:
1871:{\displaystyle {\begin{aligned}&b,a,\\&ab,\\&(ab)b,\;\;(ab)a,\\&((ab)b)b,\;\;((ab)b)a,\;\;((ab)a)a,\\&(((ab)b)b)b,\;(((ab)b)b)a,\;(((ab)b)a)a,\;(((ab)a)a)a,\;((ab)b)(ab),\;((ab)a)(ab),\\&\cdots \end{aligned}}}
5592:. In particular, this implies that no Hall word is a conjugate of another Hall word. Note again, this is exactly as it is for a Lyndon word: Lyndon words are the least of their conjugacy class; Hall words are the greatest.
1988:
833:, with parenthesis retained to show grouping. Parenthesis may be written with square brackets, so that elements of the free magma may be viewed as formal commutators. Equivalently, the free magma is the set of all
5321:
5214:
3431:
2759:
In this way, the Hall word ordering extends to a total order on the monoid. The lemmas and theorems required to provide the word-to-tree correspondence, and the unique ordering, are sketched below.
2754:
4202:
1459:
4461:
3637:
4046:
455:
5014:
4957:
3567:
3357:
4282:
3897:
5377:
3295:
3231:
4143:
2659:
388:
4628:
762:
299:
261:
2999:
4383:
3001:
otherwise. The foliage is just the string consisting of the leaves of the tree, that is, taking the tree written with commutator brackets, and erasing the commutator brackets.
1929:
3978:
51:; in fact, the Lyndon words are a special case, and almost all properties possessed by Lyndon words carry over to Hall words. Hall words are in one-to-one correspondence with
4334:
2846:
893:
587:
949:
5113:
3816:
2908:
537:
5057:
4863:
5778:
5662:
5626:
5532:
5470:
4087:
3772:
2600:
2481:
1365:
482:
322:
215:
5590:
5437:
3051:
1318:
1241:
5496:
5083:
4471:
2934:
1391:
1215:
1189:
1122:
1096:
1067:
1041:
1015:
4212:
properties of the system. The rewriting is performed by the exchange of certain pairs of consecutive Hall words in a sequence; it is given after these definitions.
3928:
3725:
3444:
The converse to the above factorization establishes the correspondence between Hall words and Hall trees. This can be stated in the following interesting form: if
3098:
2876:
2689:
2556:
2529:
1284:
989:
83:. As such, this generalizes the same process when done with the Lyndon words. Hall trees can also be used to give a total order to the elements of a group, via the
5719:
5561:
3491:
3127:
2798:
2181:
791:
4893:
4465:
Given a standard sequence with a legal drop, there are two distinct rewrite rules that create new standard sequences. One concatenates the two words in the drop:
3666:
633:
5742:
2452:
2235:
1414:
1163:
688:
505:
5690:
3462:
3167:
3147:
2414:
2391:
2371:
2347:
2327:
2307:
2279:
2255:
1977:
1957:
917:
855:
831:
811:
607:
192:
172:
1446:
665:
5800:
Because the Hall words can be placed into Hall trees, and each Hall tree can be interpreted as a commutator, the term rewrite can be used to perform the
3672:
be factored into a descending sequence of other Hall words. This implies that, given a Hall word, the corresponding tree can be uniquely identified.
2558:
can be uniquely factorized into a ascending sequence of Hall words. This is analogous to, and generalizes the better-known case of factorization with
5949:
2153:{\displaystyle M_{n}(k)\ =\ {1 \over k}\sum _{d\,|\,k}\mu \!\left({k \over d}\right)n^{d}={1 \over k}\sum _{d\,|\,k}\mu \!\left({d}\right)n^{k/d}}
1320:, may feel "backwards"; this "backwardness" is what provides the descending order required for factorization. Reversing the inequality does
5225:
5118:
690:, but this can lead to a proliferation of square brackets and commas. Keeping this in mind, one can otherwise be fluid in the notation.
1931:
elements of each distinct length. This is the beginning sequence of the necklace polynomial in two elements (described next, below).
5808:
76:
3368:
464:; the objects on the right just happen to have more structure than a magma does. To avoid the awkward typographical mess that is
2697:
5628:
is a Hall word if and only if it is larger than any of its proper right factors. This follows from the previous two points.
4148:
4392:
3578:
3983:
2494:
correspondence with the Hall trees. The total order on the Hall trees passes over to a total order on the Hall words.
399:
5815:. Note that term rewriting with Lyndon words can also be used to obtain the needed commutation for the PBW theorem.
4962:
4905:
3499:
3303:
6063:
4222:
4209:
3837:
2563:
4810:{\displaystyle (h_{1},h_{2},\ldots ,h_{n})\to (h_{1},h_{2},\ldots ,h_{i-1},h_{i+1},h_{i},h_{i+2},\ldots ,h_{n})}
1448:
to simplify notation, using parenthesis only as needed. The initial members of the Hall set are then (in order)
5326:
3236:
3172:
4092:
2608:
331:
5940:
5824:
5801:
1291:
713:
107:
95:
84:
225:, so that parenthesis must necessarily be used, when juxtaposing three or more elements. Thus, for example,
5944:
5812:
266:
228:
80:
5807:
Another very important application of the rewrite rule is to perform the commutations that appear in the
2939:
5868:
4339:
1884:
3933:
6058:
4287:
2803:
863:
542:
217:
that allows any two elements to be juxtaposed, next to each other. The juxtaposition is taken to be
925:
119:
5091:
3777:
2881:
510:
5744:
is a Hall word. This conjugate is unique, since no Hall word is a conjugate of another Hall word.
5022:
4832:
5750:
5634:
5598:
5504:
5442:
4059:
3730:
2572:
4612:{\displaystyle (h_{1},h_{2},\ldots ,h_{n})\to (h_{1},h_{2},\ldots ,h_{i}h_{i+1},\ldots ,h_{n})}
2457:
1335:
467:
307:
200:
47:, and thus provide a total order on the monoid. This is analogous to the better-known case of
5566:
5409:
4205:
3012:
2188:
1297:
1220:
5475:
5062:
2913:
1370:
1194:
1168:
1101:
1075:
1046:
1020:
994:
5980:
5844:
5794:
5387:
Hall words have a number of curious properties, many of them nearly identical to those for
3906:
3683:
3056:
2854:
2667:
2534:
2507:
2498:
1251:
956:
127:
36:
5695:
5537:
3467:
3103:
2774:
2166:
767:
8:
5665:
4872:
4866:
3645:
1980:
612:
5724:
2419:
2202:
1396:
1130:
670:
487:
87:, which is a special case of the general construction given below. It can be shown that
5836:
5675:
3447:
3152:
3132:
2399:
2376:
2356:
2332:
2312:
2292:
2264:
2240:
1962:
1942:
902:
840:
816:
796:
592:
177:
157:
115:
5963:
2184:
1419:
638:
5968:
6035:
Guy Melançon and C. Reutenauer (1989), "Lyndon words, free algebras and shuffles",
5958:
5904:
72:
3680:
The total order on Hall trees passes over to Hall words. Thus, given a Hall word
1290:
The construction and notation used below are nearly identical to that used in the
5976:
5873:
813:. The free magma is simply the set of non-associative strings in the letters of
703:
222:
218:
195:
94:
The historical development runs in reverse order from the above description. The
64:
44:
460:
The above two maps are just magma homomorphisms, in the conventional sense of a
6015:
5832:
5669:
111:
103:
5908:
6052:
5972:
28:
5852:
461:
135:
24:
5892:
5828:
5811:. A detailed discussion of the commutation is provided in the article on
5790:
5388:
2849:
2559:
2502:
834:
99:
56:
48:
40:
20:
5895:(1934), "A contribution to the theory of groups of prime-power order",
5856:
5848:
5840:
3100:
be the corresponding Hall word. Given any factorization of a Hall word
707:
151:
139:
131:
123:
68:
5316:{\displaystyle u_{1}=v_{1},\,u_{2}=v_{2},\,\ldots ,u_{k}=v_{k}\quad }
5209:{\displaystyle \quad u_{1}=v_{1},\,u_{2}=v_{2},\,\cdots ,u_{m}=v_{m}}
3436:
This and the subsequent development below are given by Guy Melançon.
5920:
W. Magnus, (1937) "Ăśber
Beziehungen zwischen höheren Kommutatoren",
3831:
5945:"A basis for free Lie rings and higher commutators in free groups"
4385:
If the sequence is a standard sequence, then the drop is termed a
5992:
X. Viennot, (1978) "Algèbres de Lie libres et monoïdes libres" ,
895:
can be constructed recursively (in increasing order) as follows:
5692:
that is a Hall word. Equivalently, there exists a factorization
635:
as needed, to disambiguate usage. Of course, one can also write
4204:
can be achieved by defining and recursively applying a simple
3169:, then there exists a factorization into Hall trees such that
75:, and can be used to perform the commutations required by the
4089:
into a concatenation of an ascending sequence of Hall words
3426:{\displaystyle y_{1}\leq y_{2}\leq \cdots \leq y_{n}\leq y.}
4048:
Note that increasing sequences of Hall words are standard.
1248:
The commutators can be ordered arbitrarily, provided that
1939:
A basic result is that the number of elements of length
5534:
is a Hall word if and only if for any factorization of
4622:
while the other permutes the two elements in the drop:
4208:. The uniqueness of the factorization follows from the
110:
based on work of Philip Hall on groups. Subsequently,
2749:{\displaystyle h_{1}\leq h_{2}\leq \cdots \leq h_{k}.}
539:. The use of parenthesis is mandatory, however, since
5753:
5727:
5698:
5678:
5637:
5601:
5569:
5540:
5507:
5478:
5445:
5412:
5329:
5228:
5121:
5094:
5065:
5025:
4965:
4908:
4875:
4835:
4631:
4474:
4395:
4342:
4290:
4225:
4197:{\displaystyle h_{1}\leq h_{2}\leq \cdots \leq h_{k}}
4151:
4095:
4062:
3986:
3936:
3909:
3840:
3780:
3733:
3686:
3648:
3581:
3502:
3470:
3450:
3371:
3306:
3239:
3175:
3155:
3135:
3106:
3059:
3015:
2942:
2916:
2884:
2857:
2806:
2777:
2700:
2670:
2611:
2575:
2537:
2510:
2460:
2422:
2402:
2379:
2359:
2335:
2315:
2295:
2267:
2243:
2205:
2169:
1991:
1965:
1945:
1887:
1457:
1422:
1399:
1373:
1338:
1300:
1254:
1223:
1197:
1171:
1133:
1104:
1078:
1049:
1023:
997:
959:
928:
905:
866:
843:
819:
799:
770:
716:
706:
subset of a free non-associative algebra, that is, a
673:
641:
615:
595:
545:
513:
490:
470:
402:
334:
310:
269:
231:
203:
180:
160:
67:
subset of a free non-associative algebra, that is, a
16:
Construction providing a total order on a free monoid
71:. In this form, the Hall trees provide a basis for
5780:is the conjugate of a power of a unique Hall word.
5772:
5736:
5713:
5684:
5656:
5620:
5584:
5555:
5526:
5490:
5464:
5431:
5371:
5315:
5208:
5107:
5077:
5051:
5008:
4951:
4887:
4869:, but at the level of Hall words. Given two words
4857:
4809:
4611:
4456:{\displaystyle h_{i+1}\leq h_{i+2},\ldots ,h_{n}.}
4455:
4377:
4328:
4276:
4196:
4137:
4081:
4040:
3972:
3922:
3891:
3810:
3766:
3719:
3660:
3631:
3561:
3485:
3456:
3425:
3351:
3289:
3225:
3161:
3141:
3121:
3092:
3045:
2993:
2928:
2902:
2870:
2840:
2792:
2748:
2683:
2653:
2594:
2550:
2523:
2475:
2446:
2408:
2385:
2365:
2341:
2321:
2301:
2273:
2249:
2229:
2175:
2152:
1971:
1951:
1923:
1870:
1440:
1408:
1385:
1359:
1312:
1278:
1235:
1209:
1183:
1157:
1116:
1090:
1061:
1035:
1009:
983:
943:
911:
887:
849:
825:
805:
785:
756:
682:
659:
627:
601:
581:
531:
499:
476:
449:
382:
316:
293:
255:
209:
186:
166:
3632:{\displaystyle f(t_{1})\leq \cdots \leq f(t_{n})}
2118:
2053:
6050:
5950:Proceedings of the American Mathematical Society
4041:{\displaystyle v_{i}\leq w_{i+1},\cdots ,w_{n}.}
3930:is either a letter, or a standard factorization
3464:is a Hall tree, and the corresponding Hall word
2602:can be written as a concatenation of Hall words
2199:Some basic definitions are useful. Given a tree
609:is a compound object, one might sometimes write
450:{\displaystyle a\bullet b\mapsto aba^{-1}b^{-1}}
5406:. That is, they cannot be written in the form
5897:Proceedings of the London Mathematical Society
5009:{\displaystyle \quad v=v_{1}v_{2}\cdots v_{n}}
4952:{\displaystyle u=u_{1}u_{2}\cdots u_{m}\quad }
3004:
1332:Consider the generating set with two elements
5789:There is a similar term rewriting system for
3562:{\displaystyle f(t)=f(t_{1})\cdots f(t_{n})}
3352:{\displaystyle x_{1},\ldots ,x_{m}>y_{1}}
2691:being totally ordered by the Hall ordering:
1351:
1339:
174:generators. This is simply a set containing
4829:A total order can be provided on the words
4277:{\displaystyle (h_{1},h_{2},\ldots ,h_{n})}
3892:{\displaystyle (w_{1},w_{2},\ldots ,w_{n})}
145:
6016:Combinatorics of Hall trees and Hall words
6010:
6008:
6006:
1817:
1780:
1743:
1706:
1669:
1598:
1597:
1569:
1568:
1515:
1514:
5962:
5372:{\displaystyle \quad u_{k+1}<v_{k+1}.}
5282:
5255:
5176:
5149:
4895:factored into ascending Hall word order,
3675:
3290:{\displaystyle v=f(y_{1})\cdots f(y_{n})}
3226:{\displaystyle u=f(x_{1})\cdots f(x_{m})}
2194:
2109:
2103:
2044:
2038:
5668:to a Hall word. That is, there exists a
4138:{\displaystyle w=h_{1}h_{2}\cdots h_{k}}
2654:{\displaystyle w=h_{1}h_{2}\cdots h_{k}}
383:{\displaystyle a\bullet b\mapsto =ab-ba}
6003:
5935:
5933:
5847:. This correspondence was motivated by
757:{\displaystyle A={a_{1},\ldots ,a_{n}}}
130:. This correspondence was motivated by
6051:
5059:a Hall word, one has an ordering that
2373:itself or an extreme right subtree of
922:The Hall set contains the generators:
4056:The unique factorization of any word
2309:itself, or a right subtree of either
294:{\displaystyle a\bullet (b\bullet c)}
256:{\displaystyle (a\bullet b)\bullet c}
5939:
5930:
5891:
5839:associated with the filtration on a
3825:
150:The setting for this article is the
5797:is accomplished with Lyndon words.
919:are given an arbitrary total order.
484:, it is conventional to just write
13:
2994:{\displaystyle f:\mapsto f(x)f(y)}
2416:is a right subtree of a Hall tree
837:with leaves marked by elements of
14:
6075:
5964:10.1090/S0002-9939-1950-0038336-7
4378:{\displaystyle h_{i}>h_{i+1}.}
4051:
3439:
1924:{\displaystyle 2,1,2,3,6,\ldots }
98:was described first, in 1934, by
4865:This is similar to the standard
3973:{\displaystyle w_{i}=u_{i}v_{i}}
1934:
764:be a set of generators, and let
304:In this way, the magma operator
59:; taken together, they form the
6037:Canadian Journal of Mathematics
6020:Journal of Combinatorial Theory
5784:
5330:
5312:
5122:
5104:
4966:
4948:
4329:{\displaystyle (h_{i},h_{i+1})}
2841:{\displaystyle f:M(A)\to A^{*}}
888:{\displaystyle H\subseteq M(A)}
582:{\displaystyle (ab)c\neq a(bc)}
106:. Hall sets were introduced by
6029:
5986:
5914:
5885:
5835:showed that they arise as the
5809:Poincaré–Birkhoff–Witt theorem
4824:
4804:
4683:
4680:
4677:
4632:
4606:
4526:
4523:
4520:
4475:
4323:
4291:
4271:
4226:
3886:
3841:
3805:
3799:
3790:
3784:
3761:
3755:
3749:
3743:
3714:
3711:
3699:
3696:
3626:
3613:
3598:
3585:
3556:
3543:
3534:
3521:
3512:
3506:
3480:
3474:
3284:
3271:
3262:
3249:
3220:
3207:
3198:
3185:
3087:
3084:
3072:
3069:
3034:
3022:
2988:
2982:
2976:
2970:
2964:
2961:
2949:
2894:
2825:
2822:
2816:
2787:
2781:
2441:
2429:
2224:
2212:
2105:
2040:
2008:
2002:
1848:
1839:
1836:
1830:
1821:
1818:
1811:
1802:
1799:
1793:
1784:
1781:
1771:
1765:
1759:
1750:
1747:
1744:
1734:
1728:
1722:
1713:
1710:
1707:
1697:
1691:
1685:
1676:
1673:
1670:
1660:
1654:
1648:
1639:
1636:
1633:
1617:
1611:
1602:
1599:
1588:
1582:
1573:
1570:
1559:
1553:
1544:
1541:
1525:
1516:
1505:
1496:
1435:
1423:
1273:
1261:
1152:
1140:
972:
960:
882:
876:
780:
774:
654:
642:
622:
616:
576:
567:
555:
546:
526:
514:
412:
359:
347:
344:
288:
276:
244:
232:
114:showed that they arise as the
79:used in the construction of a
77:Poincaré–Birkhoff–Witt theorem
1:
5879:
5823:Hall sets were introduced by
5813:universal enveloping algebras
5802:commutator collecting process
5382:
3668:. In other words, Hall words
2497:This correspondence allows a
2485:
1324:reverse this "backwardness".
1292:commutator collecting process
944:{\displaystyle A\subseteq H.}
96:commutator collecting process
85:commutator collecting process
5994:Lecture Notes in Mathematics
5108:{\displaystyle m<n\quad }
3811:{\displaystyle f(x)>f(y)}
2903:{\displaystyle f:a\mapsto a}
1979:generators) is given by the
532:{\displaystyle (a\bullet b)}
81:universal enveloping algebra
7:
5862:
5563:into non-empty words obeys
5052:{\displaystyle u_{i},v_{j}}
4858:{\displaystyle w\in A^{*}.}
3129:into two non-empty strings
3005:Factorization of Hall words
2569:More precisely, every word
693:
63:. This set is a particular
10:
6080:
5818:
5773:{\displaystyle w\in A^{*}}
5657:{\displaystyle w\in A^{*}}
5621:{\displaystyle w\in A^{*}}
5527:{\displaystyle w\in A^{*}}
5465:{\displaystyle x\in A^{*}}
5394:Other properties include:
4082:{\displaystyle w\in A^{*}}
3767:{\displaystyle w=f(x)f(y)}
3727:, it can be factorized as
2762:
2595:{\displaystyle w\in A^{*}}
1327:
393:or to a group commutator:
5831:on groups. Subsequently,
5795:factorization of a monoid
2800:is the canonical mapping
2396:A basic lemma is that if
91:coincide with Hall sets.
5855:due to Philip Hall and
4284:of Hall words is a pair
2476:{\displaystyle v\leq y.}
1360:{\displaystyle \{a,b\}.}
477:{\displaystyle \bullet }
317:{\displaystyle \bullet }
210:{\displaystyle \bullet }
146:Notational preliminaries
138:due to Philip Hall and
102:and explored in 1937 by
5909:10.1112/plms/s2-36.1.29
5585:{\displaystyle w>vu}
5432:{\displaystyle w=x^{n}}
3046:{\displaystyle t=\in H}
2564:Chen–Fox–Lyndon theorem
2283:immediate right subtree
1313:{\displaystyle v\leq y}
1236:{\displaystyle v\leq y}
793:be the free magma over
194:elements, along with a
6064:Combinatorics on words
6014:Guy Melançon, (1992) "
5869:Hall–Petresco identity
5774:
5738:
5715:
5686:
5658:
5622:
5586:
5557:
5528:
5492:
5491:{\displaystyle n>1}
5466:
5433:
5373:
5317:
5210:
5109:
5079:
5078:{\displaystyle u<v}
5053:
5010:
4953:
4889:
4859:
4811:
4613:
4457:
4379:
4330:
4278:
4198:
4139:
4083:
4042:
3974:
3924:
3893:
3820:standard factorization
3812:
3768:
3721:
3676:Standard factorization
3662:
3633:
3563:
3487:
3458:
3427:
3353:
3291:
3227:
3163:
3143:
3123:
3094:
3047:
2995:
2930:
2929:{\displaystyle a\in A}
2904:
2872:
2848:from the magma to the
2842:
2794:
2750:
2685:
2655:
2596:
2552:
2525:
2477:
2448:
2410:
2387:
2367:
2343:
2323:
2303:
2275:
2259:immediate left subtree
2251:
2231:
2195:Definitions and Lemmas
2177:
2154:
1973:
1959:in the Hall set (over
1953:
1925:
1881:Notice that there are
1872:
1442:
1410:
1387:
1386:{\displaystyle a>b}
1361:
1314:
1280:
1237:
1211:
1210:{\displaystyle v\in H}
1185:
1184:{\displaystyle u\in H}
1159:
1118:
1117:{\displaystyle y\in A}
1092:
1091:{\displaystyle x\in A}
1063:
1062:{\displaystyle x>y}
1037:
1036:{\displaystyle y\in H}
1011:
1010:{\displaystyle x\in H}
985:
945:
913:
889:
851:
827:
807:
787:
758:
684:
661:
629:
603:
583:
533:
501:
478:
451:
384:
318:
295:
257:
211:
188:
168:
6043:, No. 4, pp. 577-591.
5775:
5739:
5716:
5687:
5659:
5631:Every primitive word
5623:
5587:
5558:
5529:
5493:
5467:
5434:
5374:
5318:
5211:
5110:
5080:
5054:
5011:
4954:
4890:
4860:
4812:
4614:
4458:
4389:if one also has that
4380:
4331:
4279:
4206:term rewriting system
4199:
4140:
4084:
4043:
3975:
3925:
3923:{\displaystyle w_{i}}
3894:
3818:. This is termed the
3813:
3769:
3722:
3720:{\displaystyle w=f()}
3663:
3634:
3564:
3488:
3459:
3428:
3354:
3292:
3228:
3164:
3144:
3124:
3095:
3093:{\displaystyle w=f()}
3048:
2996:
2931:
2905:
2873:
2871:{\displaystyle A^{*}}
2843:
2795:
2751:
2686:
2684:{\displaystyle h_{j}}
2656:
2597:
2553:
2551:{\displaystyle A^{*}}
2526:
2524:{\displaystyle A^{*}}
2478:
2449:
2411:
2388:
2368:
2351:extreme right subtree
2344:
2324:
2304:
2276:
2252:
2232:
2189:Dirichlet convolution
2178:
2155:
1974:
1954:
1926:
1873:
1443:
1411:
1388:
1362:
1315:
1281:
1279:{\displaystyle y<}
1238:
1212:
1186:
1160:
1119:
1093:
1064:
1038:
1012:
986:
984:{\displaystyle \in H}
946:
914:
890:
852:
828:
808:
788:
759:
685:
662:
630:
604:
589:as already noted. If
584:
534:
502:
479:
452:
385:
319:
296:
258:
212:
189:
169:
5845:lower central series
5751:
5725:
5714:{\displaystyle w=uv}
5696:
5676:
5635:
5599:
5567:
5556:{\displaystyle w=uv}
5538:
5505:
5476:
5443:
5410:
5327:
5226:
5119:
5092:
5063:
5023:
4963:
4906:
4873:
4833:
4629:
4472:
4393:
4340:
4288:
4223:
4149:
4093:
4060:
3984:
3934:
3907:
3838:
3778:
3731:
3684:
3646:
3579:
3500:
3486:{\displaystyle f(t)}
3468:
3448:
3369:
3304:
3237:
3173:
3153:
3133:
3122:{\displaystyle w=uv}
3104:
3057:
3053:be a Hall tree, and
3013:
2940:
2914:
2882:
2855:
2804:
2793:{\displaystyle M(A)}
2775:
2698:
2668:
2664:with each Hall word
2609:
2573:
2535:
2508:
2499:monoid factorisation
2458:
2420:
2400:
2377:
2357:
2333:
2313:
2293:
2265:
2241:
2203:
2176:{\displaystyle \mu }
2167:
1989:
1963:
1943:
1885:
1455:
1420:
1397:
1371:
1336:
1298:
1252:
1221:
1195:
1169:
1131:
1102:
1076:
1047:
1021:
995:
957:
953:A formal commutator
926:
903:
864:
841:
817:
797:
786:{\displaystyle M(A)}
768:
714:
671:
639:
613:
593:
543:
511:
488:
468:
400:
332:
308:
267:
229:
201:
178:
158:
128:lower central series
118:associated with the
37:monoid factorisation
4888:{\displaystyle u,v}
4867:lexicographic order
3661:{\displaystyle n=1}
1981:necklace polynomial
628:{\displaystyle (a)}
263:is not the same as
5837:graded Lie algebra
5793:, this is how the
5770:
5737:{\displaystyle vu}
5734:
5711:
5682:
5654:
5618:
5582:
5553:
5524:
5488:
5462:
5429:
5369:
5313:
5206:
5105:
5075:
5049:
5006:
4949:
4885:
4855:
4807:
4609:
4453:
4375:
4326:
4274:
4194:
4135:
4079:
4038:
3970:
3920:
3889:
3808:
3764:
3717:
3658:
3629:
3559:
3483:
3454:
3423:
3349:
3287:
3223:
3159:
3139:
3119:
3090:
3043:
2991:
2926:
2900:
2868:
2838:
2790:
2746:
2681:
2651:
2592:
2548:
2521:
2473:
2447:{\displaystyle t=}
2444:
2406:
2383:
2363:
2339:
2319:
2299:
2271:
2247:
2230:{\displaystyle t=}
2227:
2173:
2150:
2114:
2049:
1969:
1949:
1921:
1868:
1866:
1438:
1409:{\displaystyle xy}
1406:
1383:
1357:
1310:
1276:
1233:
1207:
1181:
1158:{\displaystyle x=}
1155:
1114:
1088:
1059:
1033:
1007:
981:
941:
909:
885:
847:
823:
803:
783:
754:
683:{\displaystyle ab}
680:
657:
625:
599:
579:
529:
500:{\displaystyle ab}
497:
474:
447:
380:
314:
291:
253:
207:
184:
164:
116:graded Lie algebra
23:, in the areas of
6000:, Springer–Verlag
5827:based on work of
5685:{\displaystyle w}
3901:standard sequence
3826:Standard sequence
3457:{\displaystyle t}
3162:{\displaystyle v}
3142:{\displaystyle u}
2531:, any element of
2409:{\displaystyle v}
2386:{\displaystyle y}
2366:{\displaystyle y}
2342:{\displaystyle y}
2322:{\displaystyle x}
2302:{\displaystyle y}
2274:{\displaystyle y}
2250:{\displaystyle x}
2095:
2093:
2066:
2030:
2028:
2019:
2013:
1972:{\displaystyle n}
1952:{\displaystyle k}
912:{\displaystyle A}
850:{\displaystyle A}
826:{\displaystyle A}
806:{\displaystyle A}
602:{\displaystyle a}
187:{\displaystyle n}
167:{\displaystyle n}
73:free Lie algebras
35:provide a unique
6071:
6059:Formal languages
6044:
6033:
6027:
6026:(2) pp. 285–308.
6012:
6001:
5990:
5984:
5983:
5966:
5937:
5928:
5918:
5912:
5911:
5889:
5779:
5777:
5776:
5771:
5769:
5768:
5743:
5741:
5740:
5735:
5720:
5718:
5717:
5712:
5691:
5689:
5688:
5683:
5663:
5661:
5660:
5655:
5653:
5652:
5627:
5625:
5624:
5619:
5617:
5616:
5591:
5589:
5588:
5583:
5562:
5560:
5559:
5554:
5533:
5531:
5530:
5525:
5523:
5522:
5497:
5495:
5494:
5489:
5471:
5469:
5468:
5463:
5461:
5460:
5438:
5436:
5435:
5430:
5428:
5427:
5402:, also known as
5378:
5376:
5375:
5370:
5365:
5364:
5346:
5345:
5322:
5320:
5319:
5314:
5311:
5310:
5298:
5297:
5278:
5277:
5265:
5264:
5251:
5250:
5238:
5237:
5215:
5213:
5212:
5207:
5205:
5204:
5192:
5191:
5172:
5171:
5159:
5158:
5145:
5144:
5132:
5131:
5114:
5112:
5111:
5106:
5084:
5082:
5081:
5076:
5058:
5056:
5055:
5050:
5048:
5047:
5035:
5034:
5015:
5013:
5012:
5007:
5005:
5004:
4992:
4991:
4982:
4981:
4958:
4956:
4955:
4950:
4947:
4946:
4934:
4933:
4924:
4923:
4894:
4892:
4891:
4886:
4864:
4862:
4861:
4856:
4851:
4850:
4816:
4814:
4813:
4808:
4803:
4802:
4784:
4783:
4765:
4764:
4752:
4751:
4733:
4732:
4708:
4707:
4695:
4694:
4676:
4675:
4657:
4656:
4644:
4643:
4618:
4616:
4615:
4610:
4605:
4604:
4586:
4585:
4570:
4569:
4551:
4550:
4538:
4537:
4519:
4518:
4500:
4499:
4487:
4486:
4462:
4460:
4459:
4454:
4449:
4448:
4430:
4429:
4411:
4410:
4384:
4382:
4381:
4376:
4371:
4370:
4352:
4351:
4335:
4333:
4332:
4327:
4322:
4321:
4303:
4302:
4283:
4281:
4280:
4275:
4270:
4269:
4251:
4250:
4238:
4237:
4203:
4201:
4200:
4195:
4193:
4192:
4174:
4173:
4161:
4160:
4144:
4142:
4141:
4136:
4134:
4133:
4121:
4120:
4111:
4110:
4088:
4086:
4085:
4080:
4078:
4077:
4047:
4045:
4044:
4039:
4034:
4033:
4015:
4014:
3996:
3995:
3979:
3977:
3976:
3971:
3969:
3968:
3959:
3958:
3946:
3945:
3929:
3927:
3926:
3921:
3919:
3918:
3899:is said to be a
3898:
3896:
3895:
3890:
3885:
3884:
3866:
3865:
3853:
3852:
3817:
3815:
3814:
3809:
3773:
3771:
3770:
3765:
3726:
3724:
3723:
3718:
3667:
3665:
3664:
3659:
3638:
3636:
3635:
3630:
3625:
3624:
3597:
3596:
3568:
3566:
3565:
3560:
3555:
3554:
3533:
3532:
3492:
3490:
3489:
3484:
3463:
3461:
3460:
3455:
3432:
3430:
3429:
3424:
3413:
3412:
3394:
3393:
3381:
3380:
3358:
3356:
3355:
3350:
3348:
3347:
3335:
3334:
3316:
3315:
3296:
3294:
3293:
3288:
3283:
3282:
3261:
3260:
3232:
3230:
3229:
3224:
3219:
3218:
3197:
3196:
3168:
3166:
3165:
3160:
3148:
3146:
3145:
3140:
3128:
3126:
3125:
3120:
3099:
3097:
3096:
3091:
3052:
3050:
3049:
3044:
3000:
2998:
2997:
2992:
2935:
2933:
2932:
2927:
2909:
2907:
2906:
2901:
2877:
2875:
2874:
2869:
2867:
2866:
2847:
2845:
2844:
2839:
2837:
2836:
2799:
2797:
2796:
2791:
2755:
2753:
2752:
2747:
2742:
2741:
2723:
2722:
2710:
2709:
2690:
2688:
2687:
2682:
2680:
2679:
2660:
2658:
2657:
2652:
2650:
2649:
2637:
2636:
2627:
2626:
2601:
2599:
2598:
2593:
2591:
2590:
2562:provided by the
2557:
2555:
2554:
2549:
2547:
2546:
2530:
2528:
2527:
2522:
2520:
2519:
2482:
2480:
2479:
2474:
2453:
2451:
2450:
2445:
2415:
2413:
2412:
2407:
2392:
2390:
2389:
2384:
2372:
2370:
2369:
2364:
2348:
2346:
2345:
2340:
2328:
2326:
2325:
2320:
2308:
2306:
2305:
2300:
2280:
2278:
2277:
2272:
2261:, and likewise,
2256:
2254:
2253:
2248:
2236:
2234:
2233:
2228:
2182:
2180:
2179:
2174:
2159:
2157:
2156:
2151:
2149:
2148:
2144:
2131:
2127:
2113:
2108:
2094:
2086:
2081:
2080:
2071:
2067:
2059:
2048:
2043:
2029:
2021:
2017:
2011:
2001:
2000:
1978:
1976:
1975:
1970:
1958:
1956:
1955:
1950:
1930:
1928:
1927:
1922:
1877:
1875:
1874:
1869:
1867:
1857:
1629:
1537:
1492:
1478:
1461:
1447:
1445:
1444:
1441:{\displaystyle }
1439:
1415:
1413:
1412:
1407:
1392:
1390:
1389:
1384:
1366:
1364:
1363:
1358:
1319:
1317:
1316:
1311:
1285:
1283:
1282:
1277:
1242:
1240:
1239:
1234:
1216:
1214:
1213:
1208:
1190:
1188:
1187:
1182:
1164:
1162:
1161:
1156:
1123:
1121:
1120:
1115:
1097:
1095:
1094:
1089:
1068:
1066:
1065:
1060:
1042:
1040:
1039:
1034:
1016:
1014:
1013:
1008:
990:
988:
987:
982:
950:
948:
947:
942:
918:
916:
915:
910:
899:The elements of
894:
892:
891:
886:
856:
854:
853:
848:
832:
830:
829:
824:
812:
810:
809:
804:
792:
790:
789:
784:
763:
761:
760:
755:
753:
752:
751:
733:
732:
689:
687:
686:
681:
666:
664:
663:
660:{\displaystyle }
658:
634:
632:
631:
626:
608:
606:
605:
600:
588:
586:
585:
580:
538:
536:
535:
530:
506:
504:
503:
498:
483:
481:
480:
475:
456:
454:
453:
448:
446:
445:
433:
432:
389:
387:
386:
381:
323:
321:
320:
315:
300:
298:
297:
292:
262:
260:
259:
254:
216:
214:
213:
208:
193:
191:
190:
185:
173:
171:
170:
165:
43:. They are also
6079:
6078:
6074:
6073:
6072:
6070:
6069:
6068:
6049:
6048:
6047:
6034:
6030:
6013:
6004:
5991:
5987:
5938:
5931:
5919:
5915:
5890:
5886:
5882:
5874:Markov odometer
5865:
5821:
5787:
5764:
5760:
5752:
5749:
5748:
5726:
5723:
5722:
5697:
5694:
5693:
5677:
5674:
5673:
5648:
5644:
5636:
5633:
5632:
5612:
5608:
5600:
5597:
5596:
5568:
5565:
5564:
5539:
5536:
5535:
5518:
5514:
5506:
5503:
5502:
5477:
5474:
5473:
5456:
5452:
5444:
5441:
5440:
5423:
5419:
5411:
5408:
5407:
5398:Hall words are
5385:
5354:
5350:
5335:
5331:
5328:
5325:
5324:
5306:
5302:
5293:
5289:
5273:
5269:
5260:
5256:
5246:
5242:
5233:
5229:
5227:
5224:
5223:
5200:
5196:
5187:
5183:
5167:
5163:
5154:
5150:
5140:
5136:
5127:
5123:
5120:
5117:
5116:
5093:
5090:
5089:
5064:
5061:
5060:
5043:
5039:
5030:
5026:
5024:
5021:
5020:
5000:
4996:
4987:
4983:
4977:
4973:
4964:
4961:
4960:
4942:
4938:
4929:
4925:
4919:
4915:
4907:
4904:
4903:
4874:
4871:
4870:
4846:
4842:
4834:
4831:
4830:
4827:
4798:
4794:
4773:
4769:
4760:
4756:
4741:
4737:
4722:
4718:
4703:
4699:
4690:
4686:
4671:
4667:
4652:
4648:
4639:
4635:
4630:
4627:
4626:
4600:
4596:
4575:
4571:
4565:
4561:
4546:
4542:
4533:
4529:
4514:
4510:
4495:
4491:
4482:
4478:
4473:
4470:
4469:
4444:
4440:
4419:
4415:
4400:
4396:
4394:
4391:
4390:
4360:
4356:
4347:
4343:
4341:
4338:
4337:
4311:
4307:
4298:
4294:
4289:
4286:
4285:
4265:
4261:
4246:
4242:
4233:
4229:
4224:
4221:
4220:
4188:
4184:
4169:
4165:
4156:
4152:
4150:
4147:
4146:
4129:
4125:
4116:
4112:
4106:
4102:
4094:
4091:
4090:
4073:
4069:
4061:
4058:
4057:
4054:
4029:
4025:
4004:
4000:
3991:
3987:
3985:
3982:
3981:
3964:
3960:
3954:
3950:
3941:
3937:
3935:
3932:
3931:
3914:
3910:
3908:
3905:
3904:
3880:
3876:
3861:
3857:
3848:
3844:
3839:
3836:
3835:
3828:
3779:
3776:
3775:
3732:
3729:
3728:
3685:
3682:
3681:
3678:
3647:
3644:
3643:
3620:
3616:
3592:
3588:
3580:
3577:
3576:
3550:
3546:
3528:
3524:
3501:
3498:
3497:
3469:
3466:
3465:
3449:
3446:
3445:
3442:
3408:
3404:
3389:
3385:
3376:
3372:
3370:
3367:
3366:
3343:
3339:
3330:
3326:
3311:
3307:
3305:
3302:
3301:
3278:
3274:
3256:
3252:
3238:
3235:
3234:
3214:
3210:
3192:
3188:
3174:
3171:
3170:
3154:
3151:
3150:
3134:
3131:
3130:
3105:
3102:
3101:
3058:
3055:
3054:
3014:
3011:
3010:
3007:
2941:
2938:
2937:
2915:
2912:
2911:
2883:
2880:
2879:
2862:
2858:
2856:
2853:
2852:
2832:
2828:
2805:
2802:
2801:
2776:
2773:
2772:
2765:
2737:
2733:
2718:
2714:
2705:
2701:
2699:
2696:
2695:
2675:
2671:
2669:
2666:
2665:
2645:
2641:
2632:
2628:
2622:
2618:
2610:
2607:
2606:
2586:
2582:
2574:
2571:
2570:
2542:
2538:
2536:
2533:
2532:
2515:
2511:
2509:
2506:
2505:
2488:
2459:
2456:
2455:
2421:
2418:
2417:
2401:
2398:
2397:
2378:
2375:
2374:
2358:
2355:
2354:
2334:
2331:
2330:
2314:
2311:
2310:
2294:
2291:
2290:
2266:
2263:
2262:
2242:
2239:
2238:
2204:
2201:
2200:
2197:
2187:. The sum is a
2185:Möbius function
2183:is the classic
2168:
2165:
2164:
2140:
2136:
2132:
2123:
2119:
2104:
2099:
2085:
2076:
2072:
2058:
2054:
2039:
2034:
2020:
1996:
1992:
1990:
1987:
1986:
1964:
1961:
1960:
1944:
1941:
1940:
1937:
1886:
1883:
1882:
1865:
1864:
1855:
1854:
1627:
1626:
1535:
1534:
1490:
1489:
1476:
1475:
1458:
1456:
1453:
1452:
1421:
1418:
1417:
1398:
1395:
1394:
1372:
1369:
1368:
1337:
1334:
1333:
1330:
1299:
1296:
1295:
1253:
1250:
1249:
1222:
1219:
1218:
1196:
1193:
1192:
1170:
1167:
1166:
1132:
1129:
1128:
1103:
1100:
1099:
1077:
1074:
1073:
1048:
1045:
1044:
1022:
1019:
1018:
996:
993:
992:
991:if and only if
958:
955:
954:
927:
924:
923:
904:
901:
900:
865:
862:
861:
842:
839:
838:
818:
815:
814:
798:
795:
794:
769:
766:
765:
747:
743:
728:
724:
723:
715:
712:
711:
704:totally ordered
696:
672:
669:
668:
640:
637:
636:
614:
611:
610:
594:
591:
590:
544:
541:
540:
512:
509:
508:
489:
486:
485:
469:
466:
465:
438:
434:
425:
421:
401:
398:
397:
333:
330:
329:
309:
306:
305:
268:
265:
264:
230:
227:
226:
223:non-commutative
219:non-associative
202:
199:
198:
196:binary operator
179:
176:
175:
159:
156:
155:
148:
65:totally ordered
45:totally ordered
17:
12:
11:
5:
6077:
6067:
6066:
6061:
6046:
6045:
6028:
6002:
5985:
5957:(5): 575–581,
5941:Hall, Marshall
5929:
5913:
5883:
5881:
5878:
5877:
5876:
5871:
5864:
5861:
5851:identities in
5833:Wilhelm Magnus
5820:
5817:
5786:
5783:
5782:
5781:
5767:
5763:
5759:
5756:
5745:
5733:
5730:
5710:
5707:
5704:
5701:
5681:
5670:circular shift
5651:
5647:
5643:
5640:
5629:
5615:
5611:
5607:
5604:
5593:
5581:
5578:
5575:
5572:
5552:
5549:
5546:
5543:
5521:
5517:
5513:
5510:
5499:
5487:
5484:
5481:
5459:
5455:
5451:
5448:
5439:for some word
5426:
5422:
5418:
5415:
5384:
5381:
5380:
5379:
5368:
5363:
5360:
5357:
5353:
5349:
5344:
5341:
5338:
5334:
5309:
5305:
5301:
5296:
5292:
5288:
5285:
5281:
5276:
5272:
5268:
5263:
5259:
5254:
5249:
5245:
5241:
5236:
5232:
5217:
5216:
5203:
5199:
5195:
5190:
5186:
5182:
5179:
5175:
5170:
5166:
5162:
5157:
5153:
5148:
5143:
5139:
5135:
5130:
5126:
5103:
5100:
5097:
5074:
5071:
5068:
5046:
5042:
5038:
5033:
5029:
5017:
5016:
5003:
4999:
4995:
4990:
4986:
4980:
4976:
4972:
4969:
4945:
4941:
4937:
4932:
4928:
4922:
4918:
4914:
4911:
4884:
4881:
4878:
4854:
4849:
4845:
4841:
4838:
4826:
4823:
4818:
4817:
4806:
4801:
4797:
4793:
4790:
4787:
4782:
4779:
4776:
4772:
4768:
4763:
4759:
4755:
4750:
4747:
4744:
4740:
4736:
4731:
4728:
4725:
4721:
4717:
4714:
4711:
4706:
4702:
4698:
4693:
4689:
4685:
4682:
4679:
4674:
4670:
4666:
4663:
4660:
4655:
4651:
4647:
4642:
4638:
4634:
4620:
4619:
4608:
4603:
4599:
4595:
4592:
4589:
4584:
4581:
4578:
4574:
4568:
4564:
4560:
4557:
4554:
4549:
4545:
4541:
4536:
4532:
4528:
4525:
4522:
4517:
4513:
4509:
4506:
4503:
4498:
4494:
4490:
4485:
4481:
4477:
4452:
4447:
4443:
4439:
4436:
4433:
4428:
4425:
4422:
4418:
4414:
4409:
4406:
4403:
4399:
4374:
4369:
4366:
4363:
4359:
4355:
4350:
4346:
4325:
4320:
4317:
4314:
4310:
4306:
4301:
4297:
4293:
4273:
4268:
4264:
4260:
4257:
4254:
4249:
4245:
4241:
4236:
4232:
4228:
4219:in a sequence
4191:
4187:
4183:
4180:
4177:
4172:
4168:
4164:
4159:
4155:
4132:
4128:
4124:
4119:
4115:
4109:
4105:
4101:
4098:
4076:
4072:
4068:
4065:
4053:
4052:Term rewriting
4050:
4037:
4032:
4028:
4024:
4021:
4018:
4013:
4010:
4007:
4003:
3999:
3994:
3990:
3967:
3963:
3957:
3953:
3949:
3944:
3940:
3917:
3913:
3888:
3883:
3879:
3875:
3872:
3869:
3864:
3860:
3856:
3851:
3847:
3843:
3834:of Hall words
3827:
3824:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3763:
3760:
3757:
3754:
3751:
3748:
3745:
3742:
3739:
3736:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3677:
3674:
3657:
3654:
3651:
3640:
3639:
3628:
3623:
3619:
3615:
3612:
3609:
3606:
3603:
3600:
3595:
3591:
3587:
3584:
3570:
3569:
3558:
3553:
3549:
3545:
3542:
3539:
3536:
3531:
3527:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3493:factorizes as
3482:
3479:
3476:
3473:
3453:
3441:
3440:Correspondence
3438:
3434:
3433:
3422:
3419:
3416:
3411:
3407:
3403:
3400:
3397:
3392:
3388:
3384:
3379:
3375:
3360:
3359:
3346:
3342:
3338:
3333:
3329:
3325:
3322:
3319:
3314:
3310:
3286:
3281:
3277:
3273:
3270:
3267:
3264:
3259:
3255:
3251:
3248:
3245:
3242:
3222:
3217:
3213:
3209:
3206:
3203:
3200:
3195:
3191:
3187:
3184:
3181:
3178:
3158:
3138:
3118:
3115:
3112:
3109:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3042:
3039:
3036:
3033:
3030:
3027:
3024:
3021:
3018:
3006:
3003:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2925:
2922:
2919:
2899:
2896:
2893:
2890:
2887:
2865:
2861:
2835:
2831:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2789:
2786:
2783:
2780:
2764:
2761:
2757:
2756:
2745:
2740:
2736:
2732:
2729:
2726:
2721:
2717:
2713:
2708:
2704:
2678:
2674:
2662:
2661:
2648:
2644:
2640:
2635:
2631:
2625:
2621:
2617:
2614:
2589:
2585:
2581:
2578:
2545:
2541:
2518:
2514:
2487:
2484:
2472:
2469:
2466:
2463:
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2405:
2382:
2362:
2338:
2318:
2298:
2270:
2257:is called the
2246:
2237:, the element
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2196:
2193:
2172:
2161:
2160:
2147:
2143:
2139:
2135:
2130:
2126:
2122:
2117:
2112:
2107:
2102:
2098:
2092:
2089:
2084:
2079:
2075:
2070:
2065:
2062:
2057:
2052:
2047:
2042:
2037:
2033:
2027:
2024:
2016:
2010:
2007:
2004:
1999:
1995:
1968:
1948:
1936:
1933:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1879:
1878:
1863:
1860:
1858:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1630:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1538:
1536:
1533:
1530:
1527:
1524:
1521:
1518:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1493:
1491:
1488:
1485:
1482:
1479:
1477:
1474:
1471:
1468:
1465:
1462:
1460:
1437:
1434:
1431:
1428:
1425:
1405:
1402:
1382:
1379:
1376:
1356:
1353:
1350:
1347:
1344:
1341:
1329:
1326:
1309:
1306:
1303:
1288:
1287:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1246:
1245:
1244:
1232:
1229:
1226:
1206:
1203:
1200:
1180:
1177:
1174:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1125:
1113:
1110:
1107:
1087:
1084:
1081:
1058:
1055:
1052:
1032:
1029:
1026:
1006:
1003:
1000:
980:
977:
974:
971:
968:
965:
962:
951:
940:
937:
934:
931:
920:
908:
884:
881:
878:
875:
872:
869:
846:
822:
802:
782:
779:
776:
773:
750:
746:
742:
739:
736:
731:
727:
722:
719:
695:
692:
679:
676:
656:
653:
650:
647:
644:
624:
621:
618:
598:
578:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
528:
525:
522:
519:
516:
496:
493:
473:
458:
457:
444:
441:
437:
431:
428:
424:
420:
417:
414:
411:
408:
405:
391:
390:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
313:
290:
287:
284:
281:
278:
275:
272:
252:
249:
246:
243:
240:
237:
234:
206:
183:
163:
147:
144:
134:identities in
112:Wilhelm Magnus
104:Wilhelm Magnus
15:
9:
6:
4:
3:
2:
6076:
6065:
6062:
6060:
6057:
6056:
6054:
6042:
6038:
6032:
6025:
6021:
6017:
6011:
6009:
6007:
5999:
5995:
5989:
5982:
5978:
5974:
5970:
5965:
5960:
5956:
5952:
5951:
5946:
5942:
5936:
5934:
5926:
5923:
5917:
5910:
5906:
5902:
5898:
5894:
5888:
5884:
5875:
5872:
5870:
5867:
5866:
5860:
5858:
5854:
5850:
5846:
5843:given by the
5842:
5838:
5834:
5830:
5826:
5825:Marshall Hall
5816:
5814:
5810:
5805:
5803:
5798:
5796:
5792:
5765:
5761:
5757:
5754:
5746:
5731:
5728:
5708:
5705:
5702:
5699:
5679:
5671:
5667:
5649:
5645:
5641:
5638:
5630:
5613:
5609:
5605:
5602:
5594:
5579:
5576:
5573:
5570:
5550:
5547:
5544:
5541:
5519:
5515:
5511:
5508:
5500:
5485:
5482:
5479:
5457:
5453:
5449:
5446:
5424:
5420:
5416:
5413:
5405:
5401:
5397:
5396:
5395:
5392:
5390:
5366:
5361:
5358:
5355:
5351:
5347:
5342:
5339:
5336:
5332:
5307:
5303:
5299:
5294:
5290:
5286:
5283:
5279:
5274:
5270:
5266:
5261:
5257:
5252:
5247:
5243:
5239:
5234:
5230:
5222:
5221:
5220:
5201:
5197:
5193:
5188:
5184:
5180:
5177:
5173:
5168:
5164:
5160:
5155:
5151:
5146:
5141:
5137:
5133:
5128:
5124:
5101:
5098:
5095:
5088:
5087:
5086:
5072:
5069:
5066:
5044:
5040:
5036:
5031:
5027:
5001:
4997:
4993:
4988:
4984:
4978:
4974:
4970:
4967:
4943:
4939:
4935:
4930:
4926:
4920:
4916:
4912:
4909:
4902:
4901:
4900:
4898:
4882:
4879:
4876:
4868:
4852:
4847:
4843:
4839:
4836:
4822:
4799:
4795:
4791:
4788:
4785:
4780:
4777:
4774:
4770:
4766:
4761:
4757:
4753:
4748:
4745:
4742:
4738:
4734:
4729:
4726:
4723:
4719:
4715:
4712:
4709:
4704:
4700:
4696:
4691:
4687:
4672:
4668:
4664:
4661:
4658:
4653:
4649:
4645:
4640:
4636:
4625:
4624:
4623:
4601:
4597:
4593:
4590:
4587:
4582:
4579:
4576:
4572:
4566:
4562:
4558:
4555:
4552:
4547:
4543:
4539:
4534:
4530:
4515:
4511:
4507:
4504:
4501:
4496:
4492:
4488:
4483:
4479:
4468:
4467:
4466:
4463:
4450:
4445:
4441:
4437:
4434:
4431:
4426:
4423:
4420:
4416:
4412:
4407:
4404:
4401:
4397:
4388:
4372:
4367:
4364:
4361:
4357:
4353:
4348:
4344:
4318:
4315:
4312:
4308:
4304:
4299:
4295:
4266:
4262:
4258:
4255:
4252:
4247:
4243:
4239:
4234:
4230:
4218:
4213:
4211:
4207:
4189:
4185:
4181:
4178:
4175:
4170:
4166:
4162:
4157:
4153:
4130:
4126:
4122:
4117:
4113:
4107:
4103:
4099:
4096:
4074:
4070:
4066:
4063:
4049:
4035:
4030:
4026:
4022:
4019:
4016:
4011:
4008:
4005:
4001:
3997:
3992:
3988:
3965:
3961:
3955:
3951:
3947:
3942:
3938:
3915:
3911:
3902:
3881:
3877:
3873:
3870:
3867:
3862:
3858:
3854:
3849:
3845:
3833:
3823:
3821:
3802:
3796:
3793:
3787:
3781:
3758:
3752:
3746:
3740:
3737:
3734:
3708:
3705:
3702:
3693:
3690:
3687:
3673:
3671:
3655:
3652:
3649:
3621:
3617:
3610:
3607:
3604:
3601:
3593:
3589:
3582:
3575:
3574:
3573:
3551:
3547:
3540:
3537:
3529:
3525:
3518:
3515:
3509:
3503:
3496:
3495:
3494:
3477:
3471:
3451:
3437:
3420:
3417:
3414:
3409:
3405:
3401:
3398:
3395:
3390:
3386:
3382:
3377:
3373:
3365:
3364:
3363:
3344:
3340:
3336:
3331:
3327:
3323:
3320:
3317:
3312:
3308:
3300:
3299:
3298:
3279:
3275:
3268:
3265:
3257:
3253:
3246:
3243:
3240:
3215:
3211:
3204:
3201:
3193:
3189:
3182:
3179:
3176:
3156:
3136:
3116:
3113:
3110:
3107:
3081:
3078:
3075:
3066:
3063:
3060:
3040:
3037:
3031:
3028:
3025:
3019:
3016:
3002:
2985:
2979:
2973:
2967:
2958:
2955:
2952:
2946:
2943:
2923:
2920:
2917:
2897:
2891:
2888:
2885:
2863:
2859:
2851:
2833:
2829:
2819:
2813:
2810:
2807:
2784:
2778:
2770:
2760:
2743:
2738:
2734:
2730:
2727:
2724:
2719:
2715:
2711:
2706:
2702:
2694:
2693:
2692:
2676:
2672:
2646:
2642:
2638:
2633:
2629:
2623:
2619:
2615:
2612:
2605:
2604:
2603:
2587:
2583:
2579:
2576:
2567:
2565:
2561:
2543:
2539:
2516:
2512:
2504:
2500:
2495:
2492:
2483:
2470:
2467:
2464:
2461:
2438:
2435:
2432:
2426:
2423:
2403:
2394:
2380:
2360:
2352:
2336:
2316:
2296:
2288:
2287:right subtree
2284:
2268:
2260:
2244:
2221:
2218:
2215:
2209:
2206:
2192:
2190:
2186:
2170:
2145:
2141:
2137:
2133:
2128:
2124:
2120:
2115:
2110:
2100:
2096:
2090:
2087:
2082:
2077:
2073:
2068:
2063:
2060:
2055:
2050:
2045:
2035:
2031:
2025:
2022:
2014:
2005:
1997:
1993:
1985:
1984:
1983:
1982:
1966:
1946:
1935:Combinatorics
1932:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1861:
1859:
1851:
1845:
1842:
1833:
1827:
1824:
1814:
1808:
1805:
1796:
1790:
1787:
1777:
1774:
1768:
1762:
1756:
1753:
1740:
1737:
1731:
1725:
1719:
1716:
1703:
1700:
1694:
1688:
1682:
1679:
1666:
1663:
1657:
1651:
1645:
1642:
1631:
1623:
1620:
1614:
1608:
1605:
1594:
1591:
1585:
1579:
1576:
1565:
1562:
1556:
1550:
1547:
1539:
1531:
1528:
1522:
1519:
1511:
1508:
1502:
1499:
1494:
1486:
1483:
1480:
1472:
1469:
1466:
1463:
1451:
1450:
1449:
1432:
1429:
1426:
1403:
1400:
1380:
1377:
1374:
1354:
1348:
1345:
1342:
1325:
1323:
1307:
1304:
1301:
1293:
1286:always holds.
1270:
1267:
1264:
1258:
1255:
1247:
1230:
1227:
1224:
1204:
1201:
1198:
1178:
1175:
1172:
1149:
1146:
1143:
1137:
1134:
1126:
1111:
1108:
1105:
1085:
1082:
1079:
1071:
1070:
1056:
1053:
1050:
1030:
1027:
1024:
1004:
1001:
998:
978:
975:
969:
966:
963:
952:
938:
935:
932:
929:
921:
906:
898:
897:
896:
879:
873:
870:
867:
860:The Hall set
858:
844:
836:
820:
800:
777:
771:
748:
744:
740:
737:
734:
729:
725:
720:
717:
709:
705:
701:
691:
677:
674:
651:
648:
645:
619:
596:
573:
570:
564:
561:
558:
552:
549:
523:
520:
517:
494:
491:
471:
463:
442:
439:
435:
429:
426:
422:
418:
415:
409:
406:
403:
396:
395:
394:
377:
374:
371:
368:
365:
362:
356:
353:
350:
341:
338:
335:
328:
327:
326:
311:
302:
285:
282:
279:
273:
270:
250:
247:
241:
238:
235:
224:
220:
204:
197:
181:
161:
153:
143:
141:
137:
133:
129:
126:given by the
125:
121:
117:
113:
109:
108:Marshall Hall
105:
101:
97:
92:
90:
86:
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
29:combinatorics
26:
22:
6040:
6036:
6031:
6023:
6019:
5997:
5993:
5988:
5954:
5948:
5924:
5921:
5916:
5900:
5896:
5893:Hall, Philip
5887:
5853:group theory
5822:
5806:
5804:on a group.
5799:
5791:Lyndon words
5788:
5785:Implications
5403:
5399:
5393:
5389:Lyndon words
5386:
5218:
5018:
4896:
4828:
4819:
4621:
4464:
4386:
4216:
4214:
4055:
3900:
3829:
3819:
3679:
3669:
3641:
3571:
3443:
3435:
3361:
3008:
2768:
2766:
2758:
2663:
2568:
2560:Lyndon words
2501:: given the
2496:
2490:
2489:
2395:
2350:
2286:
2282:
2258:
2198:
2162:
1938:
1880:
1331:
1321:
1289:
859:
835:binary trees
699:
697:
667:in place of
462:homomorphism
459:
392:
303:
149:
136:group theory
93:
88:
60:
57:binary trees
55:. These are
52:
49:Lyndon words
32:
25:group theory
18:
5829:Philip Hall
5747:Every word
5085:if either
4825:Total order
2878:, given by
2850:free monoid
2771:of a magma
2503:free monoid
100:Philip Hall
89:Lazard sets
41:free monoid
21:mathematics
6053:Categories
5927:, 105-115.
5880:References
5857:Ernst Witt
5849:commutator
5841:free group
5721:such that
5383:Properties
5019:with each
4387:legal drop
4336:such that
4210:confluence
2491:Hall words
2486:Hall words
2353:is either
2289:is either
1393:and write
1098:(and thus
708:free magma
152:free magma
140:Ernst Witt
132:commutator
124:free group
120:filtration
69:free magma
53:Hall trees
33:Hall words
5973:0002-9939
5922:J. Grelle
5903:: 29–95,
5766:∗
5758:∈
5666:conjugate
5650:∗
5642:∈
5614:∗
5606:∈
5520:∗
5512:∈
5458:∗
5450:∈
5404:primitive
5284:…
5178:⋯
4994:⋯
4936:⋯
4848:∗
4840:∈
4789:…
4727:−
4713:…
4681:→
4662:…
4591:…
4556:…
4524:→
4505:…
4435:…
4413:≤
4256:…
4182:≤
4179:⋯
4176:≤
4163:≤
4123:⋯
4075:∗
4067:∈
4020:⋯
3998:≤
3871:…
3608:≤
3605:⋯
3602:≤
3538:⋯
3415:≤
3402:≤
3399:⋯
3396:≤
3383:≤
3321:…
3266:⋯
3202:⋯
3038:∈
2965:↦
2921:∈
2895:↦
2864:∗
2834:∗
2826:→
2731:≤
2728:⋯
2725:≤
2712:≤
2639:⋯
2588:∗
2580:∈
2544:∗
2517:∗
2465:≤
2171:μ
2116:μ
2097:∑
2051:μ
2032:∑
1919:…
1862:⋯
1305:≤
1228:≤
1202:∈
1176:∈
1109:∈
1083:∈
1028:∈
1002:∈
976:∈
933:⊆
871:⊆
738:…
562:≠
521:∙
472:∙
440:−
427:−
413:↦
407:∙
372:−
345:↦
339:∙
312:∙
283:∙
274:∙
248:∙
239:∙
205:∙
5943:(1950),
5863:See also
3903:if each
3832:sequence
700:Hall set
694:Hall set
61:Hall set
5981:0038336
5819:History
5595:A word
5501:A word
5400:acyclic
5219:or if
2769:foliage
2763:Foliage
2454:, then
2281:is the
1367:Define
1328:Example
1072:Either
39:of the
5979:
5971:
4145:with
3670:cannot
2163:where
2018:
2012:
710:. Let
4899:that
4897:i. e.
3980:with
3774:with
3642:then
3572:with
3297:with
3233:and
2349:. An
1165:with
1069:and:
702:is a
122:on a
5969:ISSN
5574:>
5483:>
5472:and
5348:<
5323:and
5115:and
5099:<
5070:<
4959:and
4354:>
4217:drop
3794:>
3362:and
3337:>
3149:and
3009:Let
2936:and
2910:for
2767:The
2285:. A
1416:for
1378:>
1259:<
1217:and
1191:and
1054:>
1043:and
1017:and
698:The
507:for
221:and
27:and
6024:59A
6018:",
5998:691
5959:doi
5925:177
5905:doi
5672:of
5664:is
2329:or
1322:not
1127:Or
301:.
154:in
19:In
6055::
6041:41
6039:.
6022:,
6005:^
5996:,
5977:MR
5975:,
5967:,
5953:,
5947:,
5932:^
5901:36
5899:,
5859:.
4215:A
3830:A
3822:.
2566:.
2393:.
2191:.
1124:),
857:.
142:.
31:,
5961::
5955:1
5907::
5762:A
5755:w
5732:u
5729:v
5709:v
5706:u
5703:=
5700:w
5680:w
5646:A
5639:w
5610:A
5603:w
5580:u
5577:v
5571:w
5551:v
5548:u
5545:=
5542:w
5516:A
5509:w
5498:.
5486:1
5480:n
5454:A
5447:x
5425:n
5421:x
5417:=
5414:w
5367:.
5362:1
5359:+
5356:k
5352:v
5343:1
5340:+
5337:k
5333:u
5308:k
5304:v
5300:=
5295:k
5291:u
5287:,
5280:,
5275:2
5271:v
5267:=
5262:2
5258:u
5253:,
5248:1
5244:v
5240:=
5235:1
5231:u
5202:m
5198:v
5194:=
5189:m
5185:u
5181:,
5174:,
5169:2
5165:v
5161:=
5156:2
5152:u
5147:,
5142:1
5138:v
5134:=
5129:1
5125:u
5102:n
5096:m
5073:v
5067:u
5045:j
5041:v
5037:,
5032:i
5028:u
5002:n
4998:v
4989:2
4985:v
4979:1
4975:v
4971:=
4968:v
4944:m
4940:u
4931:2
4927:u
4921:1
4917:u
4913:=
4910:u
4883:v
4880:,
4877:u
4853:.
4844:A
4837:w
4805:)
4800:n
4796:h
4792:,
4786:,
4781:2
4778:+
4775:i
4771:h
4767:,
4762:i
4758:h
4754:,
4749:1
4746:+
4743:i
4739:h
4735:,
4730:1
4724:i
4720:h
4716:,
4710:,
4705:2
4701:h
4697:,
4692:1
4688:h
4684:(
4678:)
4673:n
4669:h
4665:,
4659:,
4654:2
4650:h
4646:,
4641:1
4637:h
4633:(
4607:)
4602:n
4598:h
4594:,
4588:,
4583:1
4580:+
4577:i
4573:h
4567:i
4563:h
4559:,
4553:,
4548:2
4544:h
4540:,
4535:1
4531:h
4527:(
4521:)
4516:n
4512:h
4508:,
4502:,
4497:2
4493:h
4489:,
4484:1
4480:h
4476:(
4451:.
4446:n
4442:h
4438:,
4432:,
4427:2
4424:+
4421:i
4417:h
4408:1
4405:+
4402:i
4398:h
4373:.
4368:1
4365:+
4362:i
4358:h
4349:i
4345:h
4324:)
4319:1
4316:+
4313:i
4309:h
4305:,
4300:i
4296:h
4292:(
4272:)
4267:n
4263:h
4259:,
4253:,
4248:2
4244:h
4240:,
4235:1
4231:h
4227:(
4190:k
4186:h
4171:2
4167:h
4158:1
4154:h
4131:k
4127:h
4118:2
4114:h
4108:1
4104:h
4100:=
4097:w
4071:A
4064:w
4036:.
4031:n
4027:w
4023:,
4017:,
4012:1
4009:+
4006:i
4002:w
3993:i
3989:v
3966:i
3962:v
3956:i
3952:u
3948:=
3943:i
3939:w
3916:i
3912:w
3887:)
3882:n
3878:w
3874:,
3868:,
3863:2
3859:w
3855:,
3850:1
3846:w
3842:(
3806:)
3803:y
3800:(
3797:f
3791:)
3788:x
3785:(
3782:f
3762:)
3759:y
3756:(
3753:f
3750:)
3747:x
3744:(
3741:f
3738:=
3735:w
3715:)
3712:]
3709:y
3706:,
3703:x
3700:[
3697:(
3694:f
3691:=
3688:w
3656:1
3653:=
3650:n
3627:)
3622:n
3618:t
3614:(
3611:f
3599:)
3594:1
3590:t
3586:(
3583:f
3557:)
3552:n
3548:t
3544:(
3541:f
3535:)
3530:1
3526:t
3522:(
3519:f
3516:=
3513:)
3510:t
3507:(
3504:f
3481:)
3478:t
3475:(
3472:f
3452:t
3421:.
3418:y
3410:n
3406:y
3391:2
3387:y
3378:1
3374:y
3345:1
3341:y
3332:m
3328:x
3324:,
3318:,
3313:1
3309:x
3285:)
3280:n
3276:y
3272:(
3269:f
3263:)
3258:1
3254:y
3250:(
3247:f
3244:=
3241:v
3221:)
3216:m
3212:x
3208:(
3205:f
3199:)
3194:1
3190:x
3186:(
3183:f
3180:=
3177:u
3157:v
3137:u
3117:v
3114:u
3111:=
3108:w
3088:)
3085:]
3082:y
3079:,
3076:x
3073:[
3070:(
3067:f
3064:=
3061:w
3041:H
3035:]
3032:y
3029:,
3026:x
3023:[
3020:=
3017:t
2989:)
2986:y
2983:(
2980:f
2977:)
2974:x
2971:(
2968:f
2962:]
2959:y
2956:,
2953:x
2950:[
2947::
2944:f
2924:A
2918:a
2898:a
2892:a
2889::
2886:f
2860:A
2830:A
2823:)
2820:A
2817:(
2814:M
2811::
2808:f
2788:)
2785:A
2782:(
2779:M
2744:.
2739:k
2735:h
2720:2
2716:h
2707:1
2703:h
2677:j
2673:h
2647:k
2643:h
2634:2
2630:h
2624:1
2620:h
2616:=
2613:w
2584:A
2577:w
2540:A
2513:A
2471:.
2468:y
2462:v
2442:]
2439:y
2436:,
2433:x
2430:[
2427:=
2424:t
2404:v
2381:y
2361:y
2337:y
2317:x
2297:y
2269:y
2245:x
2225:]
2222:y
2219:,
2216:x
2213:[
2210:=
2207:t
2146:d
2142:/
2138:k
2134:n
2129:)
2125:d
2121:(
2111:k
2106:|
2101:d
2091:k
2088:1
2083:=
2078:d
2074:n
2069:)
2064:d
2061:k
2056:(
2046:k
2041:|
2036:d
2026:k
2023:1
2015:=
2009:)
2006:k
2003:(
1998:n
1994:M
1967:n
1947:k
1916:,
1913:6
1910:,
1907:3
1904:,
1901:2
1898:,
1895:1
1892:,
1889:2
1852:,
1849:)
1846:b
1843:a
1840:(
1837:)
1834:a
1831:)
1828:b
1825:a
1822:(
1819:(
1815:,
1812:)
1809:b
1806:a
1803:(
1800:)
1797:b
1794:)
1791:b
1788:a
1785:(
1782:(
1778:,
1775:a
1772:)
1769:a
1766:)
1763:a
1760:)
1757:b
1754:a
1751:(
1748:(
1745:(
1741:,
1738:a
1735:)
1732:a
1729:)
1726:b
1723:)
1720:b
1717:a
1714:(
1711:(
1708:(
1704:,
1701:a
1698:)
1695:b
1692:)
1689:b
1686:)
1683:b
1680:a
1677:(
1674:(
1671:(
1667:,
1664:b
1661:)
1658:b
1655:)
1652:b
1649:)
1646:b
1643:a
1640:(
1637:(
1634:(
1624:,
1621:a
1618:)
1615:a
1612:)
1609:b
1606:a
1603:(
1600:(
1595:,
1592:a
1589:)
1586:b
1583:)
1580:b
1577:a
1574:(
1571:(
1566:,
1563:b
1560:)
1557:b
1554:)
1551:b
1548:a
1545:(
1542:(
1532:,
1529:a
1526:)
1523:b
1520:a
1517:(
1512:,
1509:b
1506:)
1503:b
1500:a
1497:(
1487:,
1484:b
1481:a
1473:,
1470:a
1467:,
1464:b
1436:]
1433:y
1430:,
1427:x
1424:[
1404:y
1401:x
1381:b
1375:a
1355:.
1352:}
1349:b
1346:,
1343:a
1340:{
1308:y
1302:v
1274:]
1271:y
1268:,
1265:x
1262:[
1256:y
1243:.
1231:y
1225:v
1205:H
1199:v
1179:H
1173:u
1153:]
1150:v
1147:,
1144:u
1141:[
1138:=
1135:x
1112:A
1106:y
1086:A
1080:x
1057:y
1051:x
1031:H
1025:y
1005:H
999:x
979:H
973:]
970:y
967:,
964:x
961:[
939:.
936:H
930:A
907:A
883:)
880:A
877:(
874:M
868:H
845:A
821:A
801:A
781:)
778:A
775:(
772:M
749:n
745:a
741:,
735:,
730:1
726:a
721:=
718:A
678:b
675:a
655:]
652:b
649:,
646:a
643:[
623:)
620:a
617:(
597:a
577:)
574:c
571:b
568:(
565:a
559:c
556:)
553:b
550:a
547:(
527:)
524:b
518:a
515:(
495:b
492:a
443:1
436:b
430:1
423:a
419:b
416:a
410:b
404:a
378:a
375:b
369:b
366:a
363:=
360:]
357:b
354:,
351:a
348:[
342:b
336:a
289:)
286:c
280:b
277:(
271:a
251:c
245:)
242:b
236:a
233:(
182:n
162:n
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