A267433 Number of permutations of [2n] with longest increasing subsequence of length n.

1, 1, 13, 381, 17557, 1100902, 87116283, 8312317976, 927716186325, 118504614869214, 17044414451764396, 2725298085020712539, 479491040778079234419, 92050364310704637832186, 19146538134094625864605786, 4289203871330156652985437480

Offset: 0

Author

Alois P. Heinz, Jan 15 2016

Keywords

• nonn

Examples

			a(2) = 13: 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312.
		

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..74 (terms 0..55 from Alois P. Heinz)
• Mathematics StackExchange, Reference or proof for number of permutations of [2n] with longest increasing subsequence of length n, answered by David Moews, Sep 22 2023.

Crossrefs

Cf. A047874, A126065, A230341, A267436.

Programs

Maple

h:= proc(l) local n; n:= nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
    add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
a:= n-> g(n$2, [n]):
seq(a(n), n=0..20);

Mathematica

h[l_] := With[{n = Length[l]}, Total[l]! / Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], { k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := g[n, i, l] = If[n==0 || i==1, h[Join[l, Table[1, {n}]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Table[i, {j}]]], {j, 0, n/i}]]];
a[n_] := g[n, n, {n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 29 2017, translated from Maple *)

Formula

a(n) = A047874(2n,n) = A126065(2n,n).

a(n) ~ 16^n * (n-1)! / (Pi * exp(2)). - Vaclav Kotesovec, Mar 27 2016