27:
1284:
564:
to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the
122:
Algebraic combinatorics has come to be seen more expansively as an area of mathematics where the interaction of combinatorial and algebraic methods is particularly strong and significant. Thus the combinatorial topics may be
79:
The term "algebraic combinatorics" was introduced in the late 1970s. Through the early or mid-1990s, typical combinatorial objects of interest in algebraic combinatorics either admitted a lot of
444:. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions.
509:
are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite
186:
goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number
191:
505:
is not finite, because a
Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the
608:
455:, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry,
316:
Some authors exclude graphs which satisfy the definition trivially, namely those graphs which are the disjoint union of one or more equal-sized
1266:
865:
765:
913:
190:
of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the
597:
111:
556:. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite
1224:
1186:
1154:
1109:
1003:
887:
817:
588:
583:
20:
1288:
1304:
1056:
1075:
399:
1216:
1146:
411:
108:
375:
216:
satisfying certain compatibility conditions. Association schemes provide a unified approach to many topics, for example
836:
961:
171:
165:
1048:
460:
92:
124:
933:
603:
383:
578:
566:
251:
245:
88:
80:
355:
136:
60:
1258:
371:
363:
233:
217:
175:
517:
because of their regularity and simplicity. Other significant types of finite geometry are finite
437:
225:
148:
1172:
1038:
789:
751:
530:
502:
415:
209:
203:
96:
84:
907:"Applications of Matroid Theory and Combinatorial Optimization to Information and Coding Theory"
906:
1230:
1220:
1182:
1150:
1115:
1105:
1081:
1071:
1052:
1026:
999:
883:
861:
813:
761:
359:
295:
1140:
1254:
1246:
1238:
1192:
1176:
1160:
1123:
980:
948:
557:
549:
510:
498:
321:
68:
52:
897:
827:
775:
1242:
1204:
1196:
1164:
1136:
1127:
1093:
893:
823:
771:
553:
486:
480:
403:
387:
367:
229:
213:
140:
95:) or possessed a rich algebraic structure, frequently of representation theoretic origin (
1104:. Lecture Notes in Pure and Applied Mathematics. Vol. 26. Dekker. pp. 171–223.
518:
1209:
1097:
929:
537:
522:
464:
448:
395:
391:
325:
317:
144:
100:
1298:
1034:
1030:
1022:
468:
407:
379:
351:
343:
337:
267:
221:
64:
875:
545:
541:
514:
452:
441:
56:
1042:
985:
561:
48:
38:. Matroids are one of many kinds of objects studied in algebraic combinatorics.
526:
494:
390:
in 1903. Their theory was further developed by many mathematicians, including
35:
1234:
1119:
952:
1085:
845:
1283:
755:
67:
contexts and, conversely, applies combinatorial techniques to problems in
860:. Graduate Texts in Mathematics. New York: Springer-Verlag. p. 218.
490:
456:
386:, in 1900. They were then applied to the study of the symmetric group by
143:. On the algebraic side, besides group theory and representation theory,
132:
374:
groups and to study their properties. Young tableaux were introduced by
433:
427:
287:
128:
31:
26:
905:
Kashyap, Navin; Soljanin, Emina; Vontobel, Pascal (2–7 August 2009).
1181:. Progress in Mathematics. Vol. 41 (2nd ed.). Birkhäuser.
964:
Association
Schemes: Designed Experiments, Algebra and Combinatorics
757:
Association
Schemes: Designed Experiments, Algebra and Combinatorics
781:
506:
25:
1098:"Cohen–Macaulay rings, combinatorics, and simplicial complexes"
795:. School of Mathematical Sciences Shanghai Jiao Tong University
569:. Similar results hold for other kinds of finite geometries.
1102:
Ring Theory II: Proceedings of the Second
Oklahoma Conference
529:, and their higher-dimensional analogs such as higher finite
447:
Matroid theory borrows extensively from the terminology of
436:
is a structure that captures and generalizes the notion of
301:
Every two non-adjacent vertices have ÎĽ common neighbours.
228:, and the theory of association schemes generalizes the
812:. Menlo Park, CA: The Benjamin/Cummings Publishing Co.
734:
305:
A graph of this kind is sometimes said to be a srg(
1208:
835:Brouwer, Andries E.; Haemers, Willem H. (n.d.).
362:. It provides a convenient way to describe the
810:Algebraic combinatorics I: Association schemes
698:
525:, which are examples of a general type called
973:Bulletin of the American Mathematical Society
224:. In algebra, association schemes generalize
192:representation theory of the symmetric groups
103:). This period is reflected in the area 05E,
8:
722:
1068:Algebraic combinatorics on convex polytopes
1044:New Perspectives in Algebraic Combinatorics
710:
1215:. University Lecture Series. Vol. 8.
1070:. Glebe, Australia: Carslaw Publications.
686:
674:
1259:"Enumerative and Algebraic Combinatorics"
984:
638:
536:Finite geometries may be constructed via
780:. (Chapters from preliminary draft are
619:
1267:The Princeton Companion to Mathematics
662:
650:
626:
1178:Combinatorics and commutative algebra
856:Godsil, Chris; Royle, Gordon (2001).
808:Bannai, Eiichi; Ito, Tatsuro (1984).
735:Kashyap, Soljanin & Vontobel 2009
7:
174:is a specific limit of the rings of
1047:. MSRI Publications. Vol. 38.
609:Cameron–Fon-Der-Flaass IBIS theorem
1211:Gröbner bases and convex polytopes
598:Journal of Algebraic Combinatorics
112:Mathematics Subject Classification
14:
1142:Combinatorial commutative algebra
994:Zieschang, Paul-Hermann (2005b).
960:Zieschang, Paul-Hermann (2005a).
584:Combinatorial commutative algebra
560:of dimension three or greater is
21:Algebraic Combinatorics (journal)
1282:
966:by Rosemary A. Bailey, Review"
882:. New York: Chapman and Hall.
760:. Cambridge University Press.
19:For the academic journal, see
1:
1270:. Princeton University Press.
1217:American Mathematical Society
1147:Graduate Texts in Mathematics
996:Theory of association schemes
986:10.1090/S0273-0979-05-01077-3
844:. p. 101. Archived from
254:is defined as follows. Let
1149:. Vol. 227. Springer.
172:ring of symmetric functions
166:Ring of symmetric functions
1321:
1049:Cambridge University Press
699:Brouwer & Haemers n.d.
552:so constructed are called
519:Möbius or inversive planes
478:
461:combinatorial optimization
425:
412:Marcel-Paul SchĂĽtzenberger
335:
243:
201:
163:
18:
934:"Matroids you have known"
790:"Algebraic Combinatorics"
298:have λ common neighbours.
953:10.4169/193009809x469020
723:Neel & Neudauer 2009
604:Polyhedral combinatorics
51:that employs methods of
1305:Algebraic combinatorics
1289:Algebraic combinatorics
1066:Hibi, Takayuki (1992).
880:Algebraic Combinatorics
788:Bannai, Eiichi (2012).
711:Godsil & Royle 2001
590:Algebraic Combinatorics
567:non-Desarguesian planes
493:system that has only a
240:Strongly regular graphs
105:Algebraic combinatorics
89:strongly regular graphs
45:Algebraic combinatorics
858:Algebraic Graph Theory
579:Algebraic graph theory
252:strongly regular graph
246:Strongly regular graph
234:linear representations
137:partially ordered sets
114:, introduced in 1991.
39:
639:Bannai & Ito 1984
364:group representations
356:representation theory
218:combinatorial designs
176:symmetric polynomials
127:in nature or involve
61:representation theory
29:
16:Area of combinatorics
1291:at Wikimedia Commons
941:Mathematics Magazine
531:inversive geometries
384:Cambridge University
274:vertices and degree
1173:Stanley, Richard P.
1039:Stanley, Richard P.
930:Neudauer, Nancy Ann
752:Bailey, Rosemary A.
438:linear independence
290:λ and μ such that:
212:is a collection of
198:Association schemes
182:indeterminates, as
160:Symmetric functions
151:are commonly used.
149:commutative algebra
97:symmetric functions
85:association schemes
34:, derived from the
503:Euclidean geometry
416:Richard P. Stanley
286:if there are also
210:association scheme
204:Association scheme
40:
1287:Media related to
1255:Zeilberger, Doron
1023:Billera, Louis J.
867:978-0-387-95241-3
851:on 16 March 2012.
838:Spectra of Graphs
782:available on-line
767:978-0-521-82446-0
725:, pp. 26–41.
554:Galois geometries
550:projective planes
548:; the affine and
475:Finite geometries
400:G. de B. Robinson
360:Schubert calculus
354:object useful in
296:adjacent vertices
141:finite geometries
1312:
1286:
1271:
1263:
1250:
1247:Internet Archive
1214:
1205:Sturmfels, Bernd
1200:
1168:
1137:Sturmfels, Bernd
1131:
1094:Hochster, Melvin
1089:
1062:
1009:
990:
988:
970:
956:
938:
928:Neel, David L.;
924:
922:
920:
911:
901:
876:Godsil, Chris D.
871:
852:
850:
843:
831:
804:
802:
800:
794:
779:
738:
732:
726:
720:
714:
708:
702:
696:
690:
684:
678:
672:
666:
660:
654:
648:
642:
636:
630:
624:
558:projective space
540:, starting from
284:strongly regular
230:character theory
214:binary relations
155:Important topics
91:, posets with a
53:abstract algebra
1320:
1319:
1315:
1314:
1313:
1311:
1310:
1309:
1295:
1294:
1279:
1274:
1261:
1253:
1227:
1203:
1189:
1171:
1157:
1134:
1112:
1092:
1078:
1065:
1059:
1041:, eds. (1999).
1027:Björner, Anders
1021:
1017:
1015:Further reading
1012:
1006:
993:
968:
959:
936:
927:
918:
916:
909:
904:
890:
874:
868:
855:
848:
841:
834:
820:
807:
798:
796:
792:
787:
768:
750:
746:
741:
733:
729:
721:
717:
709:
705:
697:
693:
687:Zieschang 2005a
685:
681:
675:Zieschang 2005b
673:
669:
661:
657:
649:
645:
637:
633:
625:
621:
617:
575:
523:Laguerre planes
501:. The familiar
487:finite geometry
483:
481:Finite geometry
477:
430:
424:
404:Gian-Carlo Rota
388:Georg Frobenius
340:
334:
318:complete graphs
248:
242:
206:
200:
168:
162:
157:
120:
77:
24:
17:
12:
11:
5:
1318:
1316:
1308:
1307:
1297:
1296:
1293:
1292:
1278:
1277:External links
1275:
1273:
1272:
1251:
1225:
1201:
1187:
1169:
1155:
1135:Miller, Ezra;
1132:
1110:
1090:
1076:
1063:
1057:
1035:Simion, Rodica
1031:Greene, Curtis
1018:
1016:
1013:
1011:
1010:
1004:
991:
979:(2): 249–253.
957:
925:
902:
888:
872:
866:
853:
832:
818:
805:
785:
766:
747:
745:
742:
740:
739:
727:
715:
713:, p. 218.
703:
701:, p. 101.
691:
679:
667:
665:, p. 387.
655:
643:
631:
618:
616:
613:
612:
611:
606:
601:
594:
586:
581:
574:
571:
538:linear algebra
479:Main article:
476:
473:
465:network theory
449:linear algebra
426:Main article:
423:
420:
396:W. V. D. Hodge
392:Percy MacMahon
372:general linear
336:Main article:
333:
332:Young tableaux
330:
303:
302:
299:
282:is said to be
244:Main article:
241:
238:
202:Main article:
199:
196:
164:Main article:
161:
158:
156:
153:
145:lattice theory
119:
116:
101:Young tableaux
76:
73:
47:is an area of
15:
13:
10:
9:
6:
4:
3:
2:
1317:
1306:
1303:
1302:
1300:
1290:
1285:
1281:
1280:
1276:
1269:
1268:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1226:0-8218-0487-1
1222:
1218:
1213:
1212:
1206:
1202:
1198:
1194:
1190:
1188:0-8176-3836-9
1184:
1180:
1179:
1174:
1170:
1166:
1162:
1158:
1156:0-387-22356-8
1152:
1148:
1144:
1143:
1138:
1133:
1129:
1125:
1121:
1117:
1113:
1111:0-8247-6575-3
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1073:
1069:
1064:
1060:
1054:
1050:
1046:
1045:
1040:
1036:
1032:
1028:
1024:
1020:
1019:
1014:
1007:
1005:3-540-26136-2
1001:
997:
992:
987:
982:
978:
974:
967:
965:
958:
954:
950:
946:
942:
935:
931:
926:
915:
908:
903:
899:
895:
891:
889:0-412-04131-6
885:
881:
877:
873:
869:
863:
859:
854:
847:
840:
839:
833:
829:
825:
821:
819:0-8053-0490-8
815:
811:
806:
791:
786:
783:
777:
773:
769:
763:
759:
758:
753:
749:
748:
743:
736:
731:
728:
724:
719:
716:
712:
707:
704:
700:
695:
692:
688:
683:
680:
676:
671:
668:
664:
659:
656:
652:
647:
644:
640:
635:
632:
628:
623:
620:
614:
610:
607:
605:
602:
600:
599:
595:
593:
591:
587:
585:
582:
580:
577:
576:
572:
570:
568:
563:
559:
555:
551:
547:
543:
542:vector spaces
539:
534:
532:
528:
524:
520:
516:
515:affine spaces
512:
508:
504:
500:
496:
492:
488:
482:
474:
472:
470:
469:coding theory
466:
462:
458:
454:
450:
445:
443:
442:vector spaces
439:
435:
429:
421:
419:
417:
413:
409:
408:Alain Lascoux
405:
401:
397:
393:
389:
385:
381:
380:mathematician
377:
373:
369:
365:
361:
357:
353:
352:combinatorial
349:
345:
344:Young tableau
339:
338:Young tableau
331:
329:
327:
323:
319:
314:
312:
308:
300:
297:
293:
292:
291:
289:
285:
281:
277:
273:
269:
268:regular graph
265:
261:
257:
253:
247:
239:
237:
235:
231:
227:
223:
222:coding theory
219:
215:
211:
205:
197:
195:
193:
189:
185:
181:
177:
173:
167:
159:
154:
152:
150:
146:
142:
138:
134:
130:
126:
117:
115:
113:
110:
106:
102:
98:
94:
90:
86:
82:
74:
72:
70:
66:
65:combinatorial
63:, in various
62:
58:
54:
50:
46:
42:
37:
33:
28:
22:
1265:
1245:– via
1210:
1177:
1141:
1101:
1067:
1043:
998:. Springer.
995:
976:
972:
963:
947:(1): 26–41.
944:
940:
917:. Retrieved
879:
857:
846:the original
837:
809:
797:. Retrieved
756:
730:
718:
706:
694:
682:
670:
658:
646:
634:
622:
596:
589:
546:finite field
535:
484:
453:graph theory
446:
431:
376:Alfred Young
347:
341:
326:Turán graphs
320:, and their
315:
310:
306:
304:
283:
279:
275:
271:
263:
259:
255:
249:
207:
187:
183:
179:
169:
121:
104:
93:group action
78:
57:group theory
44:
43:
41:
1058:052177087-4
744:Works cited
663:Bailey 2004
651:Godsil 1993
627:Bannai 2012
527:Benz planes
322:complements
236:of groups.
125:enumerative
49:mathematics
1243:0856.13020
1197:0838.13008
1165:1066.13001
1128:0351.13009
1077:1875399046
799:30 January
562:isomorphic
511:projective
497:number of
294:Every two
81:symmetries
55:, notably
36:Fano plane
1235:907364245
1120:610144046
919:4 October
615:Citations
592:(journal)
491:geometric
368:symmetric
313:, λ, μ).
133:polytopes
107:, of the
30:The Fano
1299:Category
1257:(2008).
1207:(1996).
1175:(1996).
1139:(2005).
1096:(1977).
1086:29023080
932:(2009).
878:(1993).
754:(2004).
573:See also
457:topology
422:Matroids
348:tableaux
288:integers
129:matroids
898:1220704
828:0882540
776:2047311
544:over a
489:is any
434:matroid
428:Matroid
366:of the
350:) is a
266:) be a
75:History
69:algebra
32:matroid
1241:
1233:
1223:
1195:
1185:
1163:
1153:
1126:
1118:
1108:
1084:
1074:
1055:
1002:
896:
886:
864:
826:
816:
774:
764:
507:pixels
499:points
495:finite
346:(pl.:
324:, the
226:groups
1262:(PDF)
969:(PDF)
937:(PDF)
910:(PDF)
849:(PDF)
842:(PDF)
793:(PDF)
270:with
139:, or
118:Scope
1231:OCLC
1221:ISBN
1183:ISBN
1151:ISBN
1116:OCLC
1106:ISBN
1082:OCLC
1072:ISBN
1053:ISBN
1000:ISBN
921:2014
914:BIRS
884:ISBN
862:ISBN
814:ISBN
801:2022
762:ISBN
521:and
513:and
467:and
451:and
414:and
378:, a
370:and
358:and
220:and
170:The
147:and
59:and
1239:Zbl
1193:Zbl
1161:Zbl
1124:Zbl
981:doi
949:doi
440:in
382:at
278:.
258:= (
232:of
208:An
178:in
109:AMS
1301::
1264:.
1237:.
1229:.
1219:.
1191:.
1159:.
1145:.
1122:.
1114:.
1100:.
1080:.
1051:.
1037:;
1033:;
1029:;
1025:;
977:43
975:.
971:.
945:82
943:.
939:.
912:.
894:MR
892:.
824:MR
822:.
784:.)
772:MR
770:.
533:.
485:A
471:.
463:,
459:,
432:A
418:.
410:,
406:,
402:,
398:,
394:,
342:A
328:.
309:,
250:A
194:.
135:,
131:,
99:,
87:,
71:.
1249:.
1199:.
1167:.
1130:.
1088:.
1061:.
1008:.
989:.
983::
962:"
955:.
951::
923:.
900:.
870:.
830:.
803:.
778:.
737:.
689:.
677:.
653:.
641:.
629:.
311:k
307:v
280:G
276:k
272:v
264:E
262:,
260:V
256:G
188:n
184:n
180:n
83:(
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
