154:
148:
142:
137:
99:
131:
126:
121:
115:
110:
105:
94:
89:
84:
332:
164:
has a Durfee square of side 3 (in red) because it contains 3 parts that are ≥ 3, but does not contain 4 parts that are ≥ 4. Its Durfee symbol consists of the 2 partitions 1 and 2+1+1.
469:
415:
369:
211:
437:
It is clear from the visual definition that the Durfee square of a partition and its conjugate partition have the same size. The partitions of an integer
51:. An equivalent, but more visual, definition is that the Durfee square is the largest square that is contained within a partition's
544:
624:
599:
514:
444:
684:
591:
374:
66:
consists of the two partitions represented by the points to the right or below the Durfee square.
679:
340:
177:
485:
190:
Durfee's square is a great invention of the importance of which its author has no conception.
327:{\displaystyle P(x)=\sum _{k=0}^{\infty }{\frac {x^{k^{2}}}{\prod _{i=1}^{k}(1-x^{i})^{2}}}}
74:
The partition 4 + 3 + 3 + 2 + 1 + 1:
567:
660:
652:
8:
202:
173:
580:
620:
595:
510:
28:
656:
648:
553:
563:
52:
612:
531:
673:
181:
20:
535:
417:
represents the two sections to the right and below a Durfee square of size
640:
584:
153:
147:
141:
136:
98:
130:
125:
120:
114:
109:
104:
93:
88:
83:
558:
539:
480:
43:
is the largest number such that the partition contains at least
55:. The side-length of the Durfee square is known as the
504:
441:
contain Durfee squares with sides up to and including
529:
447:
377:
343:
214:
463:
409:
363:
326:
421:(being two partitions into parts of size at most
671:
617:James Joseph Sylvester: life and work in letters
16:Integer partition attribute, in number theory
505:Andrews, George E.; Eriksson, Kimmo (2004).
458:
448:
647:(First ed.), Oxford: Clarendon Press,
639:
464:{\displaystyle \lfloor {\sqrt {n}}\rfloor }
509:. Cambridge University Press. p. 76.
645:An introduction to the theory of numbers.
557:
619:. Oxford University Press. p. 224.
611:
425:, equivalently partitions with at most
201:The Durfee square method leads to this
672:
371:is the size of the Durfee square, and
196:
176:, a student of English mathematician
545:Electronic Journal of Combinatorics
13:
246:
14:
696:
152:
146:
140:
135:
129:
124:
119:
113:
108:
103:
97:
92:
87:
82:
172:Durfee squares are named after
633:
605:
574:
523:
498:
398:
378:
312:
292:
224:
218:
1:
491:
432:
410:{\displaystyle (1-x^{i})^{2}}
205:for the integer partitions:
35:has a Durfee square of size
7:
474:
69:
10:
701:
592:Cambridge University Press
184:in 1883, Sylvester wrote:
167:
586:Enumerative Combinatorics
364:{\displaystyle x^{k^{2}}}
643:; Wright, E. M. (1938),
613:Parshall, Karen Hunger
465:
411:
365:
328:
291:
250:
194:
178:James Joseph Sylvester
27:is an attribute of an
641:Hardy, Godfrey Harold
552:. Research Paper 32.
530:Canfield, E. Rodney;
486:Jacobi triple product
466:
412:
366:
329:
271:
230:
186:
540:"Durfee polynomials"
445:
375:
341:
212:
47:parts with values ≥
581:Stanley, Richard P.
203:generating function
197:Generating function
174:William Pitt Durfee
685:Integer partitions
507:Integer Partitions
461:
407:
361:
324:
59:of the partition.
456:
322:
180:. In a letter to
160:
159:
31:. A partition of
29:integer partition
692:
664:
663:
637:
631:
630:
609:
603:
578:
572:
571:
561:
536:Savage, Carla D.
527:
521:
520:
502:
470:
468:
467:
462:
457:
452:
416:
414:
413:
408:
406:
405:
396:
395:
370:
368:
367:
362:
360:
359:
358:
357:
333:
331:
330:
325:
323:
321:
320:
319:
310:
309:
290:
285:
269:
268:
267:
266:
252:
249:
244:
156:
150:
144:
139:
133:
128:
123:
117:
112:
107:
101:
96:
91:
86:
79:
78:
700:
699:
695:
694:
693:
691:
690:
689:
670:
669:
668:
667:
638:
634:
627:
610:
606:
579:
575:
532:Corteel, Sylvie
528:
524:
517:
503:
499:
494:
477:
451:
446:
443:
442:
435:
401:
397:
391:
387:
376:
373:
372:
353:
349:
348:
344:
342:
339:
338:
315:
311:
305:
301:
286:
275:
270:
262:
258:
257:
253:
251:
245:
234:
213:
210:
209:
199:
170:
151:
145:
134:
118:
102:
72:
53:Ferrers diagram
17:
12:
11:
5:
698:
688:
687:
682:
666:
665:
632:
625:
604:
573:
522:
515:
496:
495:
493:
490:
489:
488:
483:
476:
473:
460:
455:
450:
434:
431:
404:
400:
394:
390:
386:
383:
380:
356:
352:
347:
335:
334:
318:
314:
308:
304:
300:
297:
294:
289:
284:
281:
278:
274:
265:
261:
256:
248:
243:
240:
237:
233:
229:
226:
223:
220:
217:
198:
195:
169:
166:
162:
161:
158:
157:
71:
68:
15:
9:
6:
4:
3:
2:
697:
686:
683:
681:
680:Number theory
678:
677:
675:
662:
658:
654:
650:
646:
642:
636:
628:
626:0-19-850391-1
622:
618:
614:
608:
601:
600:0-521-56069-1
597:
593:
589:
587:
582:
577:
569:
565:
560:
559:10.37236/1370
555:
551:
547:
546:
541:
537:
533:
526:
518:
516:0-521-60090-1
512:
508:
501:
497:
487:
484:
482:
479:
478:
472:
453:
440:
430:
428:
424:
420:
402:
392:
388:
384:
381:
354:
350:
345:
316:
306:
302:
298:
295:
287:
282:
279:
276:
272:
263:
259:
254:
241:
238:
235:
231:
227:
221:
215:
208:
207:
206:
204:
193:
191:
185:
183:
182:Arthur Cayley
179:
175:
165:
155:
149:
143:
138:
132:
127:
122:
116:
111:
106:
100:
95:
90:
85:
81:
80:
77:
76:
75:
67:
65:
64:Durfee symbol
60:
58:
54:
50:
46:
42:
38:
34:
30:
26:
25:Durfee square
22:
21:number theory
644:
635:
616:
607:
585:
576:
549:
543:
525:
506:
500:
438:
436:
426:
422:
418:
336:
200:
189:
187:
171:
163:
73:
63:
61:
56:
48:
44:
40:
36:
32:
24:
18:
674:Categories
661:0020.29201
653:64.0093.03
590:, p. 289.
588:, Volume 2
492:References
433:Properties
459:⌋
449:⌊
385:−
299:−
273:∏
247:∞
232:∑
615:(1998).
538:(1998).
475:See also
429:parts).
70:Examples
583:(1999)
568:1631751
481:h-index
168:History
659:
651:
623:
598:
566:
513:
337:where
621:ISBN
596:ISBN
511:ISBN
62:The
57:rank
23:, a
657:Zbl
649:JFM
594:.
554:doi
39:if
19:In
676::
655:,
602:.
564:MR
562:.
548:.
542:.
534:;
471:.
629:.
570:.
556::
550:5
519:.
454:n
439:n
427:k
423:k
419:k
403:2
399:)
393:i
389:x
382:1
379:(
355:2
351:k
346:x
317:2
313:)
307:i
303:x
296:1
293:(
288:k
283:1
280:=
277:i
264:2
260:k
255:x
242:0
239:=
236:k
228:=
225:)
222:x
219:(
216:P
192:"
188:"
49:s
45:s
41:s
37:s
33:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
