A342573 - OEIS

%I #26 Feb 02 2022 23:34:11

%S 1,2,24,5184,39813120,17915904000000,702142910300160000000,

%T 3330690501757390081228800000000,

%U 2534703826002712645182542460223488000000000,395940866122425193243875570782668457763038822400000000000

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

%C 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.

%C This sequence also counts the sets of n permutations of size n such that the i-th element of each of them is the same.

%C a(n) is a subsequence of A001013: products of factorial numbers.

%H Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, <a href="https://arxiv.org/abs/2201.00645">Sequences of the Stable Matching Problem</a>, arXiv:2201.00645 [math.HO], 2021.

%F a(n) = n*(n-1)!^n = n*A091868(n-1).

%e 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.

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

%Y Cf. A001013, A091868.

%K nonn

%O 1,2

%A _Tanya Khovanova_ and MIT PRIMES STEP Senior group, Mar 27 2021