This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A351409 #53 Feb 13 2022 09:23:20
%S A351409 1,8,3888,764411904,214990848000000000,224634374557469245440000000000,
%T A351409 1880461634768804771224006806208512000000000000,
%U A351409 240091793104790737576620139562796649430329798636339200000000000000,813675117804798213250391541747787241264315446434692481270971279693253181440000000000000000
%N A351409 a(n) = n*(n!)^(2*n-2).
%C A351409 a(n) is the number of reduced Stable Marriage Problem instances of order n. In the SMP, relabeling men or women has no effect on the number of stable matchings. So the men and women can be relabeled to normalize the order of man #1's rankings (with woman #1 as his first choice and woman n as his last choice), and to similarly normalize the order of woman #1's rankings, except for her ranking of man #1. This reduces the number of possible instances by a factor of n!(n-1)! (A010790 with shifted offset), from (n!)^(2n) (A185141) to a(n). This reduction is directly analogous to the identical reduction from latin squares (A002860) to reduced latin squares (A000315), and can be directly applied to the Latin Stable Marriage Problem (A351413). As with reduced latin squares, some further reduction is possible analogous to row/column reduced latin squares (A123234).
%C A351409 It is tempting to aim for a reduction of (n!)^2 by simultaneously normalizing all of man #1 and woman #1's preferences, but that isn't possible unless man #1 and woman #1 happen to be mutual first choices.
%C A351409 Applying this reduction to A344669 reduces A344669(2) and A344669(4) to 1, demonstrating that these maximal instances arising in A005154 are unique up to participant relabeling. It raises the question of which other values of n make A344669(n) reducible to 1.
%H A351409 Andrew Howroyd, <a href="/A351409/b351409.txt">Table of n, a(n) for n = 1..20</a>
%H A351409 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, [Section 7, Symmetries].
%F A351409 a(n) = A185141(n) / A010790(n-1).
%t A351409 a[n_] := n*(n!)^(2*n - 2); Table[a[n], {n, 1, 9}] (* _Robert P. P. McKone_, Feb 12 2022 *)
%o A351409 (PARI) a(n) = n*(n!)^(2*n-2) \\ _Andrew Howroyd_, Feb 12 2022
%Y A351409 Cf. A185141, A000315, A351413, A344669.
%K A351409 nonn
%O A351409 1,2
%A A351409 _Dan Eilers_, Feb 11 2022