

A342573


The number of ordered ntuples consisting of n permutations (not necessarily distinct) such that the first element of each of them is the same.


6



1, 2, 24, 5184, 39813120, 17915904000000, 702142910300160000000, 3330690501757390081228800000000, 2534703826002712645182542460223488000000000, 395940866122425193243875570782668457763038822400000000000
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OFFSET

1,2


COMMENTS

This is related to the stable marriage problem, as this counts the preference profiles for n men trying to marry n women when all of them prefer the same woman.
This sequence also counts the sets of n permutations of size n such that the ith element of each of them is the same.
a(n) is a subsequence of A001013: products of factorial numbers.


LINKS

Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021.


FORMULA

a(n) = n*(n1)!^n = n*A091868(n1).


EXAMPLE

When n=3, we have 3 ways to fix the first element, and the remaining elements in each permutation can be in any order, yielding (3  1)! possible ways of ordering the rest of each permutation, so there are 3 * (2!)^3 = 24 sets of permutations.


MATHEMATICA

Table[n (n  1)!^n, {n, 10}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



