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 A351580 #28 Jun 02 2023 15:56:13
%S A351580 1,2,21,2600,9078630,1634935320144,22831938997720867560,
%T A351580 34390564970975286088924022400,
%U A351580 7457911916650283082000186530740981347120,300682790088737748950725540713718365319268411170195200,2830053444386286847574443631356044745870287426798365860653876609636480
%N A351580 a(n) is the number of multisets of size n-1 consisting of permutations of n elements.
%C A351580 a(n) is the number of reduced men's ranking tables in the stable marriage problem of order n. In the SMP (as noted in A351409), relabeling men or women has no effect on the number of stable matchings. So the 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 then the men except man #1 can be relabeled to normalize the lexicographic order of those men's rankings. Since man #1's rankings end up fixed in natural order, they do not contribute to the number of possibilities, leaving n! multichoose (n-1) ways to arrange the rankings of the other n-1 men.
%C A351580 The number of unreduced men's ranking tables is given by A036740. Relabeling just the women reduces this to A134366. Alternately, relabeling just the men reduces A036740 to A344690. Relabeling both men and women reduces the men's relabeling reduction, A344690, by a factor of (n!+n-1)/n to a(n).
%C A351580 It might be tempting to try to reduce A344690 by a factor of n!, but that doesn't work because not all of man #1's rankings are equally likely after relabeling all the men to give man #1 the lexicographically least rankings.
%C A351580 There is room for further relabeling reduction from a(n), given by A263921. The reduction from a(n) to A263921 is analogous to the reduction from reduced latin squares, A000315, to A123234.
%C A351580 Each of the a(n) reduced men's ranking tables can be combined with the A036740 possible unreduced women's ranking tables to form complete instances, but these instances have more possibilities than A351409. For example, a(3)*A036740(3)=21*216=4536 > A351409(3)=3888. However, fewer possibilities result from using A263921 in place of a(n), although the men's ranking tables of A263921 may not be as straightforward to generate. With A263921(3)=10, 10*216=2160 < 3888.
%H A351580 Alois P. Heinz, <a href="/A351580/b351580.txt">Table of n, a(n) for n = 1..31</a>
%H A351580 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multiset#Counting_multisets">Counting multisets</a>
%F A351580 a(n) = binomial(n! + n - 2, n - 1).
%F A351580 a(n) = n*A344690(n)/A030495(n-1).
%F A351580 a(n) = A344690*n/(n! + n - 1).
%F A351580 a(n) = A071919(n-1,n!). - _Alois P. Heinz_, Feb 16 2022
%e A351580 Starting with the following men's ranking table of order 3, where row k represents man k's rankings, the 1 in the 2nd position of row 3 means that man #3 ranks woman #2 as his 1st choice.
%e A351580 213
%e A351580 321
%e A351580 213
%e A351580 Step 1: reorder columns so row 1 is in natural order:
%e A351580 123
%e A351580 231
%e A351580 123
%e A351580 Step 2: reorder rows 2 to n so rows are in lexical order:
%e A351580 123
%e A351580 123
%e A351580 231
%e A351580 a(3)=21 because there are 1+2+3+4+5+6 = 21 possibilities for the last two rows in lexical order, with 3!=6 possible permutations for each row.
%e A351580 The 21 tables for a(3) are the following:
%e A351580 123 123 123 123 123 123 123
%e A351580 123 123 123 123 123 123 132
%e A351580 123 132 213 231 312 321 132
%e A351580 .
%e A351580 123 123 123 123 123 123 123
%e A351580 132 132 132 132 213 213 213
%e A351580 213 231 312 321 213 231 312
%e A351580 .
%e A351580 123 123 123 123 123 123 123
%e A351580 213 231 231 231 312 312 321
%e A351580 321 231 312 321 312 321 321
%t A351580 Table[Binomial[n!+n-2,n-1],{n,15}] (* _Harvey P. Dale_, Jun 02 2023 *)
%o A351580 (PARI) a(n) = binomial(n! + n - 2, n - 1) \\ _Andrew Howroyd_, Feb 13 2022
%Y A351580 Cf. A036740, A071919, A134366, A344690, A263921, A351409, A000315, A185141, A030495.
%K A351580 nonn
%O A351580 1,2
%A A351580 _Dan Eilers_, Feb 13 2022
%E A351580 Erroneous Mathematica program deleted by _N. J. A. Sloane_, Jun 02 2023