***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001938 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. Let $\pi$ be a permutation of cycle type $\mu$. A transitive monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ is a tuple of $r = n + \ell(\mu) - 2$ transpositions $$ (a_1, b_1),\dots,(a_r, b_r) $$ with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, such that the subgroup of $\mathfrak S_n$ generated by the transpositions acts transitively on $\{1,\dots,n\}$ and hose product, in this order, is $\pi$. ----------------------------------------------------------------------------- References: [1] Goulden, I. P., Guay-Paquet, M., Novak, J. Monotone Hurwitz numbers in genus zero [[MathSciNet:3095005]] ----------------------------------------------------------------------------- Code: def statistic(mu): return Hurwitz(mu) / mu.conjugacy_class_size() def Hurwitz(alpha): alpha = Partition(alpha) d = alpha.size() r = factorial(d) / prod(factorial(m) for m in alpha.to_exp()) * prod(binomial(2*p, p) for p in alpha) k = len(alpha) - 3 if k >= 0: return r * rising_factorial(2*d + 1, k) return r / rising_factorial(2*d + k + 1, -k) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 1 [1,1] => 1 [3] => 2 [2,1] => 4 [1,1,1] => 8 [4] => 5 [3,1] => 15 [2,2] => 18 [2,1,1] => 48 [1,1,1,1] => 144 [5] => 14 [4,1] => 56 [3,2] => 72 [3,1,1] => 240 [2,2,1] => 288 [2,1,1,1] => 1056 [1,1,1,1,1] => 4224 [6] => 42 [5,1] => 210 [4,2] => 280 [4,1,1] => 1120 [3,3] => 300 [3,2,1] => 1440 [3,1,1,1] => 6240 [2,2,2] => 1728 [2,2,1,1] => 7488 [2,1,1,1,1] => 34944 [1,1,1,1,1,1] => 174720 [7] => 132 [6,1] => 792 [5,2] => 1080 [5,1,1] => 5040 [4,3] => 1200 [4,2,1] => 6720 [4,1,1,1] => 33600 [3,3,1] => 7200 [3,2,2] => 8640 [3,2,1,1] => 43200 [3,1,1,1,1] => 230400 [2,2,2,1] => 51840 [2,2,1,1,1] => 276480 [2,1,1,1,1,1] => 1566720 [1,1,1,1,1,1,1] => 9400320 [8] => 429 [7,1] => 3003 [6,2] => 4158 [6,1,1] => 22176 [5,3] => 4725 [5,2,1] => 30240 [5,1,1,1] => 171360 [4,4] => 4900 [4,3,1] => 33600 [4,2,2] => 40320 [4,2,1,1] => 228480 [4,1,1,1,1] => 1370880 [3,3,2] => 43200 [3,3,1,1] => 244800 [3,2,2,1] => 293760 [3,2,1,1,1] => 1762560 [3,1,1,1,1,1] => 11162880 [2,2,2,2] => 352512 [2,2,2,1,1] => 2115072 [2,2,1,1,1,1] => 13395456 [2,1,1,1,1,1,1] => 89303040 [1,1,1,1,1,1,1,1] => 625121280 [9] => 1430 [8,1] => 11440 [7,2] => 16016 [7,1,1] => 96096 [6,3] => 18480 [6,2,1] => 133056 [6,1,1,1] => 842688 [5,4] => 19600 [5,3,1] => 151200 [5,2,2] => 181440 [5,2,1,1] => 1149120 [5,1,1,1,1] => 7660800 [4,4,1] => 156800 [4,3,2] => 201600 [4,3,1,1] => 1276800 [4,2,2,1] => 1532160 [4,2,1,1,1] => 10214400 [4,1,1,1,1,1] => 71500800 [3,3,3] => 216000 [3,3,2,1] => 1641600 [3,3,1,1,1] => 10944000 [3,2,2,2] => 1969920 [3,2,2,1,1] => 13132800 [3,2,1,1,1,1] => 91929600 [3,1,1,1,1,1,1] => 674150400 [2,2,2,2,1] => 15759360 [2,2,2,1,1,1] => 110315520 [2,2,1,1,1,1,1] => 808980480 [10] => 4862 [9,1] => 43758 [8,2] => 61776 [8,1,1] => 411840 [7,3] => 72072 [7,2,1] => 576576 [7,1,1,1] => 4036032 [6,4] => 77616 [6,3,1] => 665280 [6,2,2] => 798336 [6,2,1,1] => 5588352 [6,1,1,1,1] => 40981248 [5,5] => 79380 [5,4,1] => 705600 [5,3,2] => 907200 [5,3,1,1] => 6350400 [5,2,2,1] => 7620480 [5,2,1,1,1] => 55883520 [5,1,1,1,1,1] => 428440320 [4,4,2] => 940800 [4,4,1,1] => 6585600 [4,3,3] => 1008000 [4,3,2,1] => 8467200 [4,3,1,1,1] => 62092800 [4,2,2,2] => 10160640 [4,2,2,1,1] => 74511360 [4,2,1,1,1,1] => 571253760 [3,3,3,1] => 9072000 [3,3,2,2] => 10886400 [3,3,2,1,1] => 79833600 [3,3,1,1,1,1] => 612057600 [3,2,2,2,1] => 95800320 [3,2,2,1,1,1] => 734469120 [2,2,2,2,2] => 114960384 [2,2,2,2,1,1] => 881362944 [11] => 16796 [10,1] => 167960 [9,2] => 238680 [9,1,1] => 1750320 [8,3] => 280800 [8,2,1] => 2471040 [8,1,1,1] => 18944640 [7,4] => 305760 [7,3,1] => 2882880 [7,2,2] => 3459456 [7,2,1,1] => 26522496 [7,1,1,1,1] => 212179968 [6,5] => 317520 [6,4,1] => 3104640 [6,3,2] => 3991680 [6,3,1,1] => 30602880 [6,2,2,1] => 36723456 [6,2,1,1,1] => 293787648 [5,5,1] => 3175200 [5,4,2] => 4233600 [5,4,1,1] => 32457600 [5,3,3] => 4536000 [5,3,2,1] => 41731200 [5,3,1,1,1] => 333849600 [5,2,2,2] => 50077440 [5,2,2,1,1] => 400619520 [4,4,3] => 4704000 [4,4,2,1] => 43276800 [4,4,1,1,1] => 346214400 [4,3,3,1] => 46368000 [4,3,2,2] => 55641600 [4,3,2,1,1] => 445132800 [4,2,2,2,1] => 534159360 [3,3,3,2] => 59616000 [3,3,3,1,1] => 476928000 [3,3,2,2,1] => 572313600 [3,2,2,2,2] => 686776320 [12] => 58786 [11,1] => 646646 [10,2] => 923780 [10,1,1] => 7390240 [9,3] => 1093950 [9,2,1] => 10501920 [9,1,1,1] => 87516000 [8,4] => 1201200 [8,3,1] => 12355200 [8,2,2] => 14826240 [8,2,1,1] => 123552000 [8,1,1,1,1] => 1070784000 [7,5] => 1261260 [7,4,1] => 13453440 [7,3,2] => 17297280 [7,3,1,1] => 144144000 [7,2,2,1] => 172972800 [7,2,1,1,1] => 1499097600 [6,6] => 1280664 [6,5,1] => 13970880 [6,4,2] => 18627840 [6,4,1,1] => 155232000 [6,3,3] => 19958400 [6,3,2,1] => 199584000 [6,3,1,1,1] => 1729728000 [6,2,2,2] => 239500800 [6,2,2,1,1] => 2075673600 [5,5,2] => 19051200 [5,5,1,1] => 158760000 [5,4,3] => 21168000 [5,4,2,1] => 211680000 [5,4,1,1,1] => 1834560000 [5,3,3,1] => 226800000 [5,3,2,2] => 272160000 [4,4,4] => 21952000 [4,4,3,1] => 235200000 [4,4,2,2] => 282240000 [4,3,3,2] => 302400000 [3,3,3,3] => 324000000 [13] => 208012 [12,1] => 2496144 [11,2] => 3581424 [11,1,1] => 31039008 [10,3] => 4263600 [10,2,1] => 44341440 [10,1,1,1] => 399072960 [9,4] => 4712400 [9,3,1] => 52509600 [9,2,2] => 63011520 [9,2,1,1] => 567103680 [8,5] => 4989600 [8,4,1] => 57657600 [8,3,2] => 74131200 [8,3,1,1] => 667180800 [8,2,2,1] => 800616960 [7,6] => 5122656 [7,5,1] => 60540480 [7,4,2] => 80720640 [7,4,1,1] => 726485760 [7,3,3] => 86486400 [7,3,2,1] => 934053120 [7,2,2,2] => 1120863744 [6,6,1] => 61471872 [6,5,2] => 83825280 [6,5,1,1] => 754427520 [6,4,3] => 93139200 [6,4,2,1] => 1005903360 [6,3,3,1] => 1077753600 [6,3,2,2] => 1293304320 [5,5,3] => 95256000 [5,5,2,1] => 1028764800 [5,4,4] => 98784000 [5,4,3,1] => 1143072000 [5,4,2,2] => 1371686400 [5,3,3,2] => 1469664000 [4,4,4,1] => 1185408000 [4,4,3,2] => 1524096000 [4,3,3,3] => 1632960000 [14] => 742900 [13,1] => 9657700 [12,2] => 13907088 [12,1,1] => 129799488 [11,3] => 16628040 [11,2,1] => 186234048 [11,1,1,1] => 1800262464 [10,4] => 18475600 [10,3,1] => 221707200 [10,2,2] => 266048640 [9,5] => 19691100 [9,4,1] => 245044800 [9,3,2] => 315057600 [8,6] => 20386080 [8,5,1] => 259459200 [8,4,2] => 345945600 [8,3,3] => 370656000 [7,7] => 20612592 [7,6,1] => 266378112 [7,5,2] => 363242880 [7,4,3] => 403603200 [6,6,2] => 368831232 [6,5,3] => 419126400 [6,4,4] => 434649600 [5,5,4] => 444528000 [15] => 2674440 [14,1] => 37442160 [13,2] => 54083120 [13,1,1] => 540831200 [12,3] => 64899744 [12,2,1] => 778796928 [11,4] => 72424352 [11,3,1] => 931170240 [11,2,2] => 1117404288 [10,5] => 77597520 [10,4,1] => 1034633600 [10,3,2] => 1330243200 [9,6] => 80864784 [9,5,1] => 1102701600 [9,4,2] => 1470268800 [9,3,3] => 1575288000 [8,7] => 82450368 [8,6,1] => 1141620480 [8,5,2] => 1556755200 [8,4,3] => 1729728000 [7,7,1] => 1154305152 [7,6,2] => 1598268672 [7,5,3] => 1816214400 [7,4,4] => 1883481600 [6,6,3] => 1844156160 [6,5,4] => 1955923200 [5,5,5] => 2000376000 [16] => 9694845 [15,1] => 145422675 [14,2] => 210612150 [13,3] => 253514625 [12,4] => 283936380 [11,5] => 305540235 [10,6] => 320089770 [9,7] => 328513185 [8,8] => 331273800 [17] => 35357670 [16,1] => 565722720 [15,2] => 821210400 [14,3] => 991116000 [13,4] => 1113476000 [12,5] => 1202554080 [11,6] => 1265296032 [10,7] => 1305464160 [9,8] => 1325095200 ----------------------------------------------------------------------------- Created: Jan 01, 2024 at 23:37 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Aug 05, 2024 at 22:53 by Martin Rubey