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Number of permutations of n distinct letters (ABCD...) each of which appears 5 times and having no fixed points.
3

%I #27 Aug 22 2022 11:31:18

%S 1,0,1,2252,44127009,2671644472544,413723943299025265,

%T 142244957218019486750604,97613348575755314842878968833,

%U 123144020654635535717072991038686496,267585539125011749129687143446506422964961,950060633410906693026597892010516600171358115820

%N Number of permutations of n distinct letters (ABCD...) each of which appears 5 times and having no fixed points.

%C a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 6 pure options. - _Raimundas Vidunas_, Jan 22 2014

%H Shalosh B. Ekhad, Christoph Koutschan, and Doron Zeilberger, <a href="https://arxiv.org/abs/2101.10147">There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts]</a>, arXiv:2101.10147 [math.CO], 2021.

%H R. D. McKelvey and A. McLennan, <a href="https://doi.org/10.1006/jeth.1996.2214">The maximal number of regular totally mixed Nash equilibria</a>, J. Economic Theory, 72 (1997), 411-425.

%H R. Vidunas, <a href="http://arxiv.org/abs/1401.5400">MacMahon's master theorem and totally mixed Nash equilibria</a>, arXiv preprint arXiv:1401.5400 [math.CO], 2014-2016.

%H Vidunas, Raimundas <a href="https://doi.org/10.1007/s00026-017-0344-2">Counting derangements and Nash equilibria</a> Ann. Comb. 21, No. 1, 131-152 (2017).

%e "1"

%e "0", 0, 0, 0, 0, 1

%e "1", 0, 25, 0, 100, 0, 100, 0, 25, 0, 1

%e "2252", 15150, 48600, 99350, 144150, 156753, 131000, 87075, 45000, 19300, 6000, 1800, 250, 75, 0, 1

%e "44127009", 274314600, 822998550, 1583402400, 2189652825, 2311947008, 1932997200, 1310330400, 731686550, 340071600, 132480756, 43364000, 11973150, 2760000, 541600, 84000, 12225, 1000, 150, 0, 1

%e etc.

%p p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); for n from 0 to 8 do seq(coeff(f(t, n, 5), t, m)/5!^n, m=0..5*n); od;

%t p[x_, k_] := k!^2 Sum[x^j/((k - j)!^2 j!), {j, 0, k}];

%t R[x_, n_, k_] := p[x, k]^n;

%t f[t_, n_, k_] := Sum[Coefficient[R[x, n, k], x, j] (t - 1)^j (n k - j)!, {j, 0, n k}];

%t Reap[For[n = 0, n <= 11, n++, Sow[Table[Coefficient[f[t, n, 5], t, m]/5!^n, {m, 0, 5n}]]]][[2, 1]][[All, 1]] (* _Jean-François Alcover_, Aug 19 2018, from Maple *)

%Y Cf. A059062.

%K nonn

%O 0,4

%A _Zerinvary Lajos_, Nov 07 2006

%E More terms from _Alois P. Heinz_, Sep 27 2015