*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* 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