A359856 Number of permutations of [1..n] which are indecomposable by direct and skew sums.

1, 1, 0, 0, 2, 22, 202, 1854, 17866, 183806, 2029850, 24081006, 306486314, 4175102110, 60708557626, 939518187726, 15430666746826, 268214861561726, 4921023843969242, 95066628485598126, 1929291834938927210, 41042364285004263262, 913409469123533445754, 21227246586149632119438

Offset: 0

Author

Ludovic Schwob, Jan 16 2023

Keywords

• nonn

Examples

			The only permutations of [1..4] which are indecomposable by direct and skew sums are 2413 and 3142.
		

Links

• Wikipedia, Skew and direct sums of permutations

Crossrefs

Cf. A003319, A111111.

Programs

Mathematica

nmax = 20; CoefficientList[Series[2*(2 - 1/Sum[k!*x^k, {k, 0, nmax}]) - Sum[k!*x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 19 2023 *)

PARI

seq(n)={my(p=sum(k=0, n, k!*x^k, O(x*x^n))); Vec(2*(2 - 1/p) - p)} \\ Andrew Howroyd, Jan 16 2023

Formula

G.f.: 2*(2 - 1/F(x)) - F(x) where F(x) = Sum_{k>=0} k!*x^k.

G.f.: S(F(x)) - 2*F(x)^2 - F(x) + x + 1 where S(x) is the g.f. of A111111 and F(x) = Sum_{k>=1} k!*x^k.

a(n) ~ n! * (1 - 4/n - 2/n^2 - 10/n^3 - 64/n^4 - 506/n^5 - 4762/n^6 - 51824/n^7 - 638678/n^8 - 8777898/n^9 - 132990772/n^10 - ...). - Vaclav Kotesovec, Jan 19 2023