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

0,5

Links

Table of n, a(n) for n=0..23.

Wikipedia, Skew and direct sums of permutations

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A003319, A111111.

Sequence in context: A043037 A058441 A255043 * A366919 A308313 A304024

Adjacent sequences: A359852 A359853 A359854 * A359857 A359858 A359859

Keyword

nonn

Author

Ludovic Schwob, Jan 16 2023

Status

approved