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A342573 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. 6
1, 2, 24, 5184, 39813120, 17915904000000, 702142910300160000000, 3330690501757390081228800000000, 2534703826002712645182542460223488000000000, 395940866122425193243875570782668457763038822400000000000 (list; graph; refs; listen; history; text; internal format)
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 i-th element of each of them is the same.
a(n) is a subsequence of A001013: products of factorial numbers.
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.
a(n) = n*(n-1)!^n = n*A091868(n-1).
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.
Table[n (n - 1)!^n, {n, 10}]
Sequence in context: A159907 A242484 A088912 * A203465 A055462 A088600
Tanya Khovanova and MIT PRIMES STEP Senior group, Mar 27 2021

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Last modified May 18 08:32 EDT 2024. Contains 372618 sequences. (Running on oeis4.)