A321313 Number of permutations of [n] with equal lengths of the longest increasing subsequence and the longest decreasing subsequence.

1, 0, 4, 4, 36, 256, 1282, 9864, 99976, 970528, 9702848, 113092200, 1500063930, 20985500212, 305177475748, 4733232671056, 79461918315024, 1427464201289584, 26955955609799728, 531536672155429792, 10980840178654738496, 238597651836121062824, 5446220581860028853936

Offset: 1

Author

Alois P. Heinz, Nov 03 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..80
• Wikipedia, Longest increasing subsequence

Crossrefs

Column k=0 of A321316.

Cf. A000142, A003316, A321314, A321315.

Programs

Maple

h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(j>
    l[k], 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
f:= l-> `if`(l[1]=nops(l), h(l)^2, 0):
g:= (n, i, l)-> `if`(n=0 or i=1, f([l[], 1$n]),
     g(n, i-1, l) +g(n-i, min(i, n-i), [l[], i])):
a:= n-> g(n$2, []):
seq(a(n), n=1..23);

Mathematica

h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[j > l[[k]], 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
f[l_] := If[l[[1]] == Length[l], h[l]^2, 0];
g[n_, i_, l_] := If[n == 0 || i == 1, f[Join[l, Table[1, {n}]]], g[n, i - 1, l] + g[n - i, Min[i, n - i], Append[l, i]]];
a[n_] := g[n, n, {}];
Array[a, 25] (* Jean-François Alcover, Aug 31 2021, after Alois P. Heinz *)

Formula

a(n) = n! - 2 * A321314(n).

a(n) = A321315(n) - A321314(n).

a(n) = A321316(n,0).