108:
20:
1563:
1458:
1417:
1376:
1335:
1294:
1253:
1212:
1171:
1130:
1089:
170:
1630:
1019:
987:
955:
923:
891:
859:
827:
795:
763:
731:
699:
667:
611:
579:
1592:
1048:
547:
521:
495:
469:
443:
417:
391:
305:
232:
1650:
1510:
1478:
631:
365:
345:
325:
198:
129:
101:
81:
61:
41:
266:
234:
terms and whose edges connect two bracketings that can be obtained from one another either by moving a pair of brackets using
1771:
254:
239:
307:, the vertices of the permutoassociahedron can be represented by bracketing all the permutations of three terms
1594:
terms and the number of all possible bracketings of any such permutation. The former number is equal to the
1709:(1993). "The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation".
258:
1814:
1518:
523:, and each of them admits two bracketings (obtained from one another by associativity). For instance,
1737:
270:
1769:
Baralić, Djordje; Ivanović, Jelena; Petrić, Zoran (December 2019). "A simple permutoassociahedron".
1819:
1780:
1422:
1381:
1340:
1299:
1258:
1217:
1176:
1135:
1094:
1053:
262:
134:
1790:
1751:
1718:
1706:
1600:
250:
992:
960:
928:
896:
864:
832:
800:
768:
736:
704:
672:
640:
584:
552:
1481:
43:
and the correspondence between its vertices and the bracketed permutations of three terms
1571:
1027:
526:
500:
474:
448:
422:
396:
370:
284:
211:
1653:
1635:
1495:
1463:
616:
350:
330:
310:
183:
114:
107:
86:
66:
46:
26:
1808:
1722:
1681:
1676:
235:
19:
1742:
1686:
205:
1794:
1755:
1664:
1660:
246:
1595:
634:
1091:
is adjacent to exactly three other vertices of the permutoassociahedron:
201:
172:. Three of these facets are quadrilaterals and the fourth is a pentagon.
1667:
with two different kinds of flips (associativity and transpositions).
1568:
vertices. This is the product between the number of permutations of
1785:
106:
18:
1255:
via associativity and the third via a transposition. The vertex
269:. It was constructed as a convex polytope by Victor Reiner and
253:
who noted that this structure appears implicitly in Mac Lane's
1659:
By its description in terms of bracketed permutations, the
1460:, via a transposition. This illustrates that, in dimension
242:
two consecutive terms that are not separated by a bracket.
111:
The four facets of the permutoassociahedron of dimension
1638:
1603:
1574:
1521:
1498:
1466:
1425:
1384:
1343:
1302:
1261:
1220:
1179:
1138:
1097:
1056:
1030:
995:
963:
931:
899:
867:
835:
803:
771:
739:
707:
675:
643:
619:
587:
555:
529:
503:
477:
451:
425:
399:
373:
353:
333:
313:
287:
214:
186:
137:
117:
89:
69:
49:
29:
1378:, are reached via associativity, and the other two,
204:
whose vertices correspond to the bracketings of the
1644:
1624:
1586:
1557:
1504:
1472:
1452:
1411:
1370:
1329:
1288:
1247:
1206:
1165:
1124:
1083:
1042:
1013:
981:
949:
917:
885:
853:
821:
789:
757:
725:
693:
661:
625:
605:
573:
541:
515:
489:
463:
437:
411:
385:
359:
339:
319:
299:
226:
192:
164:
123:
95:
75:
55:
35:
1549:
1531:
245:The permutoassociahedron was first defined as a
1480:and above, the permutoassociahedron is not a
8:
1296:is adjacent to four vertices. Two of them,
1214:. The first two vertices are reached from
1784:
1637:
1602:
1573:
1548:
1530:
1528:
1520:
1497:
1465:
1424:
1383:
1342:
1301:
1260:
1219:
1178:
1137:
1096:
1055:
1029:
994:
962:
930:
898:
866:
834:
802:
770:
738:
706:
674:
642:
633:-dimensional permutoassociahedron is the
618:
586:
554:
528:
502:
476:
450:
424:
398:
372:
352:
332:
312:
286:
213:
185:
136:
116:
88:
68:
48:
28:
1698:
1512:-dimensional permutoassociahedron has
23:The permutoassociahedron of dimension
7:
1711:Journal of Pure and Applied Algebra
367:. There are six such permutations,
1535:
257:theorem for symmetric and braided
14:
1663:of the permutoassociahedron is a
1740:(1994). "Coxeter-associahedra".
1558:{\displaystyle n!{2n \choose n}}
267:Knizhnik–Zamolodchikov equations
1616:
1604:
1447:
1438:
1435:
1426:
1406:
1397:
1394:
1385:
1365:
1362:
1353:
1347:
1321:
1315:
1306:
1303:
1283:
1274:
1271:
1262:
1239:
1233:
1224:
1221:
1198:
1192:
1183:
1180:
1157:
1154:
1145:
1139:
1119:
1110:
1107:
1098:
1075:
1069:
1060:
1057:
1005:
996:
973:
964:
944:
935:
912:
903:
877:
868:
845:
836:
816:
807:
784:
775:
749:
740:
717:
708:
688:
679:
656:
647:
600:
591:
565:
556:
159:
150:
147:
138:
1:
1723:10.1016/0022-4049(93)90049-Y
1836:
1795:10.1016/j.disc.2019.07.007
1756:10.1112/S0025579300007452
1453:{\displaystyle (ab)(dc)}
1412:{\displaystyle (ba)(cd)}
1371:{\displaystyle a(b(cd))}
1330:{\displaystyle ((ab)c)d}
1289:{\displaystyle (ab)(cd)}
1248:{\displaystyle ((ab)c)d}
1207:{\displaystyle ((ba)c)d}
1166:{\displaystyle (a(bc))d}
1125:{\displaystyle (ab)(cd)}
1084:{\displaystyle ((ab)c)d}
165:{\displaystyle (ab)(cd)}
1646:
1626:
1625:{\displaystyle (n+1)!}
1588:
1559:
1506:
1474:
1454:
1413:
1372:
1331:
1290:
1249:
1208:
1167:
1126:
1085:
1044:
1015:
983:
951:
919:
887:
855:
823:
791:
759:
727:
695:
663:
627:
607:
575:
543:
517:
491:
465:
439:
413:
387:
361:
341:
321:
301:
228:
194:
173:
166:
125:
104:
97:
77:
57:
37:
1647:
1632:and the later is the
1627:
1589:
1560:
1507:
1475:
1455:
1414:
1373:
1332:
1291:
1250:
1209:
1168:
1127:
1086:
1045:
1016:
1014:{\displaystyle (ab)c}
984:
982:{\displaystyle (ba)c}
952:
950:{\displaystyle b(ac)}
920:
918:{\displaystyle b(ca)}
888:
886:{\displaystyle (bc)a}
856:
854:{\displaystyle (cb)a}
824:
822:{\displaystyle c(ba)}
792:
790:{\displaystyle c(ab)}
760:
758:{\displaystyle (ca)b}
728:
726:{\displaystyle (ac)b}
696:
694:{\displaystyle a(cb)}
664:
662:{\displaystyle a(bc)}
628:
608:
606:{\displaystyle a(bc)}
576:
574:{\displaystyle (ab)c}
544:
518:
492:
466:
440:
414:
388:
362:
342:
322:
302:
229:
195:
167:
126:
110:
98:
78:
58:
38:
22:
1772:Discrete Mathematics
1707:Kapranov, Mikhail M.
1636:
1601:
1572:
1519:
1496:
1464:
1423:
1382:
1341:
1300:
1259:
1218:
1177:
1136:
1095:
1054:
1028:
993:
961:
929:
897:
865:
833:
801:
769:
737:
705:
673:
641:
617:
585:
553:
549:can be bracketed as
527:
501:
475:
449:
423:
397:
371:
351:
331:
311:
285:
212:
184:
178:permutoassociahedron
176:In mathematics, the
135:
115:
87:
67:
47:
27:
1587:{\displaystyle n+1}
1043:{\displaystyle n=3}
542:{\displaystyle abc}
516:{\displaystyle cba}
490:{\displaystyle cab}
464:{\displaystyle bca}
438:{\displaystyle bac}
412:{\displaystyle acb}
386:{\displaystyle abc}
300:{\displaystyle n=2}
227:{\displaystyle n+1}
1738:Ziegler, Günter M.
1642:
1622:
1584:
1555:
1502:
1470:
1450:
1409:
1368:
1327:
1286:
1245:
1204:
1163:
1122:
1081:
1040:
1011:
979:
947:
915:
883:
851:
819:
787:
755:
723:
691:
659:
623:
603:
571:
539:
513:
487:
461:
435:
409:
383:
357:
337:
317:
297:
224:
190:
174:
162:
131:that share vertex
121:
105:
93:
73:
53:
33:
1645:{\displaystyle n}
1547:
1505:{\displaystyle n}
1473:{\displaystyle 3}
626:{\displaystyle 2}
360:{\displaystyle c}
340:{\displaystyle b}
320:{\displaystyle a}
271:Günter M. Ziegler
263:Vladimir Drinfeld
193:{\displaystyle n}
124:{\displaystyle 3}
96:{\displaystyle c}
76:{\displaystyle b}
56:{\displaystyle a}
36:{\displaystyle 2}
1827:
1799:
1798:
1788:
1766:
1760:
1759:
1736:Reiner, Victor;
1733:
1727:
1726:
1703:
1651:
1649:
1648:
1643:
1631:
1629:
1628:
1623:
1593:
1591:
1590:
1585:
1564:
1562:
1561:
1556:
1554:
1553:
1552:
1543:
1534:
1511:
1509:
1508:
1503:
1479:
1477:
1476:
1471:
1459:
1457:
1456:
1451:
1418:
1416:
1415:
1410:
1377:
1375:
1374:
1369:
1336:
1334:
1333:
1328:
1295:
1293:
1292:
1287:
1254:
1252:
1251:
1246:
1213:
1211:
1210:
1205:
1172:
1170:
1169:
1164:
1131:
1129:
1128:
1123:
1090:
1088:
1087:
1082:
1049:
1047:
1046:
1041:
1020:
1018:
1017:
1012:
988:
986:
985:
980:
956:
954:
953:
948:
924:
922:
921:
916:
892:
890:
889:
884:
860:
858:
857:
852:
828:
826:
825:
820:
796:
794:
793:
788:
764:
762:
761:
756:
732:
730:
729:
724:
700:
698:
697:
692:
668:
666:
665:
660:
632:
630:
629:
624:
612:
610:
609:
604:
580:
578:
577:
572:
548:
546:
545:
540:
522:
520:
519:
514:
496:
494:
493:
488:
470:
468:
467:
462:
444:
442:
441:
436:
418:
416:
415:
410:
392:
390:
389:
384:
366:
364:
363:
358:
346:
344:
343:
338:
326:
324:
323:
318:
306:
304:
303:
298:
251:Mikhail Kapranov
233:
231:
230:
225:
199:
197:
196:
191:
171:
169:
168:
163:
130:
128:
127:
122:
102:
100:
99:
94:
82:
80:
79:
74:
62:
60:
59:
54:
42:
40:
39:
34:
1835:
1834:
1830:
1829:
1828:
1826:
1825:
1824:
1805:
1804:
1803:
1802:
1768:
1767:
1763:
1735:
1734:
1730:
1705:
1704:
1700:
1695:
1673:
1634:
1633:
1599:
1598:
1570:
1569:
1536:
1529:
1517:
1516:
1494:
1493:
1490:
1482:simple polytope
1462:
1461:
1421:
1420:
1380:
1379:
1339:
1338:
1298:
1297:
1257:
1256:
1216:
1215:
1175:
1174:
1134:
1133:
1093:
1092:
1052:
1051:
1026:
1025:
991:
990:
959:
958:
927:
926:
895:
894:
863:
862:
831:
830:
799:
798:
767:
766:
735:
734:
703:
702:
671:
670:
639:
638:
615:
614:
583:
582:
551:
550:
525:
524:
499:
498:
473:
472:
447:
446:
421:
420:
395:
394:
369:
368:
349:
348:
329:
328:
309:
308:
283:
282:
279:
265:'s work on the
210:
209:
182:
181:
133:
132:
113:
112:
85:
84:
65:
64:
45:
44:
25:
24:
17:
12:
11:
5:
1833:
1831:
1823:
1822:
1817:
1807:
1806:
1801:
1800:
1779:(12): 111591.
1761:
1750:(2): 364–393.
1728:
1717:(2): 119–142.
1697:
1696:
1694:
1691:
1690:
1689:
1684:
1679:
1672:
1669:
1654:Catalan number
1641:
1621:
1618:
1615:
1612:
1609:
1606:
1583:
1580:
1577:
1566:
1565:
1551:
1546:
1542:
1539:
1533:
1527:
1524:
1501:
1489:
1486:
1469:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1039:
1036:
1033:
1010:
1007:
1004:
1001:
998:
978:
975:
972:
969:
966:
946:
943:
940:
937:
934:
914:
911:
908:
905:
902:
882:
879:
876:
873:
870:
850:
847:
844:
841:
838:
818:
815:
812:
809:
806:
786:
783:
780:
777:
774:
754:
751:
748:
745:
742:
722:
719:
716:
713:
710:
690:
687:
684:
681:
678:
658:
655:
652:
649:
646:
637:with vertices
622:
602:
599:
596:
593:
590:
570:
567:
564:
561:
558:
538:
535:
532:
512:
509:
506:
486:
483:
480:
460:
457:
454:
434:
431:
428:
408:
405:
402:
382:
379:
376:
356:
336:
316:
296:
293:
290:
278:
275:
261:as well as in
223:
220:
217:
189:
161:
158:
155:
152:
149:
146:
143:
140:
120:
92:
72:
52:
32:
15:
13:
10:
9:
6:
4:
3:
2:
1832:
1821:
1818:
1816:
1813:
1812:
1810:
1796:
1792:
1787:
1782:
1778:
1774:
1773:
1765:
1762:
1757:
1753:
1749:
1745:
1744:
1739:
1732:
1729:
1724:
1720:
1716:
1712:
1708:
1702:
1699:
1692:
1688:
1685:
1683:
1682:Associahedron
1680:
1678:
1677:Permutohedron
1675:
1674:
1670:
1668:
1666:
1662:
1657:
1655:
1639:
1619:
1613:
1610:
1607:
1597:
1581:
1578:
1575:
1544:
1540:
1537:
1525:
1522:
1515:
1514:
1513:
1499:
1487:
1485:
1483:
1467:
1444:
1441:
1432:
1429:
1403:
1400:
1391:
1388:
1359:
1356:
1350:
1344:
1324:
1318:
1312:
1309:
1280:
1277:
1268:
1265:
1242:
1236:
1230:
1227:
1201:
1195:
1189:
1186:
1160:
1151:
1148:
1142:
1116:
1113:
1104:
1101:
1078:
1072:
1066:
1063:
1050:, the vertex
1037:
1034:
1031:
1022:
1008:
1002:
999:
976:
970:
967:
941:
938:
932:
909:
906:
900:
880:
874:
871:
848:
842:
839:
813:
810:
804:
781:
778:
772:
752:
746:
743:
720:
714:
711:
685:
682:
676:
653:
650:
644:
636:
620:
613:. Hence, the
597:
594:
588:
568:
562:
559:
536:
533:
530:
510:
507:
504:
484:
481:
478:
458:
455:
452:
432:
429:
426:
406:
403:
400:
380:
377:
374:
354:
334:
314:
294:
291:
288:
276:
274:
272:
268:
264:
260:
256:
252:
248:
243:
241:
237:
236:associativity
221:
218:
215:
207:
203:
200:-dimensional
187:
179:
156:
153:
144:
141:
118:
109:
90:
70:
50:
30:
21:
1815:Permutations
1776:
1770:
1764:
1747:
1741:
1731:
1714:
1710:
1701:
1658:
1567:
1491:
1023:
280:
244:
206:permutations
177:
175:
1743:Mathematika
1687:Cyclohedron
240:transposing
1809:Categories
1786:1708.02482
1693:References
1665:flip graph
1661:1-skeleton
1488:Properties
259:categories
247:CW complex
1820:Polytopes
1596:factorial
635:dodecagon
255:coherence
1671:See also
277:Examples
202:polytope
16:Polytope
1173:, and
989:, and
581:or as
497:, and
347:, and
238:or by
180:is an
83:, and
1781:arXiv
1024:When
281:When
1492:The
1419:and
1337:and
1791:doi
1777:342
1752:doi
1719:doi
1652:th
249:by
208:of
1811::
1789:.
1775:.
1748:41
1746:.
1715:85
1713:.
1656:.
1484:.
1132:,
1021:.
957:,
925:,
893:,
861:,
829:,
797:,
765:,
733:,
701:,
669:,
471:,
445:,
419:,
393:,
327:,
273:.
63:,
1797:.
1793::
1783::
1758:.
1754::
1725:.
1721::
1640:n
1620:!
1617:)
1614:1
1611:+
1608:n
1605:(
1582:1
1579:+
1576:n
1550:)
1545:n
1541:n
1538:2
1532:(
1526:!
1523:n
1500:n
1468:3
1448:)
1445:c
1442:d
1439:(
1436:)
1433:b
1430:a
1427:(
1407:)
1404:d
1401:c
1398:(
1395:)
1392:a
1389:b
1386:(
1366:)
1363:)
1360:d
1357:c
1354:(
1351:b
1348:(
1345:a
1325:d
1322:)
1319:c
1316:)
1313:b
1310:a
1307:(
1304:(
1284:)
1281:d
1278:c
1275:(
1272:)
1269:b
1266:a
1263:(
1243:d
1240:)
1237:c
1234:)
1231:b
1228:a
1225:(
1222:(
1202:d
1199:)
1196:c
1193:)
1190:a
1187:b
1184:(
1181:(
1161:d
1158:)
1155:)
1152:c
1149:b
1146:(
1143:a
1140:(
1120:)
1117:d
1114:c
1111:(
1108:)
1105:b
1102:a
1099:(
1079:d
1076:)
1073:c
1070:)
1067:b
1064:a
1061:(
1058:(
1038:3
1035:=
1032:n
1009:c
1006:)
1003:b
1000:a
997:(
977:c
974:)
971:a
968:b
965:(
945:)
942:c
939:a
936:(
933:b
913:)
910:a
907:c
904:(
901:b
881:a
878:)
875:c
872:b
869:(
849:a
846:)
843:b
840:c
837:(
817:)
814:a
811:b
808:(
805:c
785:)
782:b
779:a
776:(
773:c
753:b
750:)
747:a
744:c
741:(
721:b
718:)
715:c
712:a
709:(
689:)
686:b
683:c
680:(
677:a
657:)
654:c
651:b
648:(
645:a
621:2
601:)
598:c
595:b
592:(
589:a
569:c
566:)
563:b
560:a
557:(
537:c
534:b
531:a
511:a
508:b
505:c
485:b
482:a
479:c
459:a
456:c
453:b
433:c
430:a
427:b
407:b
404:c
401:a
381:c
378:b
375:a
355:c
335:b
315:a
295:2
292:=
289:n
222:1
219:+
216:n
188:n
160:)
157:d
154:c
151:(
148:)
145:b
142:a
139:(
119:3
103:.
91:c
71:b
51:a
31:2
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