A321314 Number of permutations of [n] where the length of the longest increasing subsequence is larger than the length of the longest decreasing subsequence.

0, 1, 1, 10, 42, 232, 1879, 15228, 131452, 1329136, 15106976, 182954700, 2363478435, 33096395494, 501248446126, 8094778608472, 138112754890488, 2487454752219208, 47344572399516136, 950682668010605104, 20055050996527350752, 442701537970743308588, 10202898078512473893032

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

Cf. A000142, A003316, A321313, A321315, A321316.

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] `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) = Sum_{k=1..n-1} A321316(n,k).

a(n) = (n! - A321313(n))/2.

a(n) = A321315(n) - A321313(n).