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%I #18 Apr 26 2024 09:24:53
%S 1,1,13,381,17557,1100902,87116283,8312317976,927716186325,
%T 118504614869214,17044414451764396,2725298085020712539,
%U 479491040778079234419,92050364310704637832186,19146538134094625864605786,4289203871330156652985437480
%N Number of permutations of [2n] with longest increasing subsequence of length n.
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A267433/b267433.txt">Table of n, a(n) for n = 0..74</a> (terms 0..55 from Alois P. Heinz)
%H Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/4771623/reference-or-proof-for-number-of-permutations-of-2n-with-longest-increasing-su">Reference or proof for number of permutations of [2n] with longest increasing subsequence of length n</a>, answered by David Moews, Sep 22 2023.
%F a(n) = A047874(2n,n) = A126065(2n,n).
%F a(n) ~ 16^n * (n-1)! / (Pi * exp(2)). - _Vaclav Kotesovec_, Mar 27 2016
%e a(2) = 13: 1432, 2143, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312.
%p h:= proc(l) local n; n:= nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
%p add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
%p g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
%p add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
%p a:= n-> g(n$2, [n]):
%p seq(a(n), n=0..20);
%t 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}]];
%t 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}]]];
%t a[n_] := g[n, n, {n}];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Mar 29 2017, translated from Maple *)
%Y Cf. A047874, A126065, A230341, A267436.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jan 15 2016
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