edit this statistic or download as text // json
Identifier
Values
=>
Cc0002;cc-rep
[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
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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)
Created
Jan 01, 2024 at 23:37 by Martin Rubey
Updated
Aug 05, 2024 at 22:53 by Martin Rubey