A123297 Number of permutations of n distinct letters (ABCD...) each of which appears 5 times and having no fixed points.
1, 0, 1, 2252, 44127009, 2671644472544, 413723943299025265, 142244957218019486750604, 97613348575755314842878968833, 123144020654635535717072991038686496, 267585539125011749129687143446506422964961, 950060633410906693026597892010516600171358115820
Offset: 0
Keywords
Examples
"1" "0", 0, 0, 0, 0, 1 "1", 0, 25, 0, 100, 0, 100, 0, 25, 0, 1 "2252", 15150, 48600, 99350, 144150, 156753, 131000, 87075, 45000, 19300, 6000, 1800, 250, 75, 0, 1 "44127009", 274314600, 822998550, 1583402400, 2189652825, 2311947008, 1932997200, 1310330400, 731686550, 340071600, 132480756, 43364000, 11973150, 2760000, 541600, 84000, 12225, 1000, 150, 0, 1 etc.
Links
- Shalosh B. Ekhad, Christoph Koutschan, and Doron Zeilberger, There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts], arXiv:2101.10147 [math.CO], 2021.
- R. D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72 (1997), 411-425.
- R. Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arXiv preprint arXiv:1401.5400 [math.CO], 2014-2016.
- Vidunas, Raimundas Counting derangements and Nash equilibria Ann. Comb. 21, No. 1, 131-152 (2017).
Crossrefs
Cf. A059062.
Programs
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Maple
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;
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Mathematica
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[Coefficient[R[x, n, k], x, j] (t - 1)^j (n k - j)!, {j, 0, n k}]; 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 *)
Extensions
More terms from Alois P. Heinz, Sep 27 2015
Comments