A246073 - OEIS

%I #18 Nov 13 2020 01:23:04

%S 1,1,10,108,6672,109200,45007920,983324160,665546434560,

%T 60174422501760,32648180513760000,4656975300322329600,

%U 13859947861644771532800,1193599114668580293273600,1257285172911535450293811200,766119340152013216053484800000

%N Number of permutations p on [2n] satisfying p^n(i) = i for all i in [n].

%C Conjecture: Lim inf n->infinity a(n) / (((n-1)!)^2 * 4^(n-1) / sqrt(n)) = 1.128... . - _Vaclav Kotesovec_, Aug 14 2014

%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A246073/b246073.txt">Table of n, a(n) for n = 0..233</a> (first 100 terms from Alois P. Heinz)

%H Vaclav Kotesovec, <a href="/A246073/a246073.jpg">Graph - asymptotic</a>

%F a(n) = A246072(2n,n).

%e a(2) = 10: (1,2,3,4), (1,2,4,3), (1,3,2,4), (1,4,3,2), (2,1,3,4), (2,1,4,3), (3,2,1,4), (3,4,1,2), (4,2,3,1), (4,3,2,1).

%e a(3) = 108: (1,2,3,4,5,6), (1,2,3,4,6,5), (1,2,3,5,4,6), ... (6,4,2,3,1,5), (6,5,1,2,4,3), (6,5,2,1,3,4).

%p with(numtheory): with(combinat): M:=multinomial:

%p b:= proc(n, k, p) local l, g; l, g:= sort([divisors(p)[]]),

%p       proc(k, m, i, t) option remember; local d, j; d:= l[i];

%p         `if`(i=1, m!, add(M(k, k-(d-t)*j, (d-t)$j)/j!*

%p          (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,

%p         `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),

%p         `if`(t=0, [][], m/t))))

%p       end; g(k, n-k, nops(l), 0)

%p     end:

%p a:= n-> `if`(n=0, 1, b(2*n, n, n)):

%p seq(a(n), n=0..20);

%t multinomial[n_, k_List] := n!/Times @@ (k!); M = multinomial;

%t b[n_, k_, p_] := b[n, k, p] = Module[{l, g}, l = Sort[Divisors[p]]; g[k0_, m_, i_, t_] := g[k0, m, i, t] = Module[{d}, d = l[[i]]; If[i == 1, m!, Sum[M[k0, Join[{k0 - (d - t) j}, Table[d - t, {j}]]]/j! (d - 1)!^j M[m, Join[{m - t j}, Table[t, {j}]]] If[d - t == 1, g[k0 - (d - t) j, m - t j, i - 1, 0], g[k0 - (d - t) j, m - t j, i, t + 1]], {j, 0, Min[k0/(d - t), If[t == 0, Infinity, m/t]]}]]]; g[k, n - k, Length[l], 0]];

%t a[n_] := If[n == 0, 1, b[2n, n, n]];

%t a /@ Range[0, 20] (* _Jean-François Alcover_, Nov 13 2020, after _Alois P. Heinz_ *)

%Y Main diagonal of A246072.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 12 2014