0,2

a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {1>2>3>4} of length 5. That is, the number of length n+1 permutations having no subsequences of length 5 in which the element in position 1 is larger than the element in position 2, which in turn is larger than the element in position 3, and that element is larger than the element in position 4. - Sergey Kitaev, Dec 13 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Sucharita Biswas, Umesh Shankar, and Sivaramakrishnan Sivasubramanian, Ordinal and disjoint sums of partially ordered patterns, arXiv:2510.18761 [math.CO], 2025. See p. 17.

Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.

Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.

a(n) = (n+1)*A005802(n), where A005802(n) = number of permutations in S_n with longest increasing subsequence of length <= 3.

a(n) = Sum_{k=0..n} C(2k,k)*C(n,k)*C(n+1,k)/(k+1)^2.

Recurrence: (n+2)^2*a(n) = (n+1)*(7*n+2)*a(n-1) + 3*(n-2)*(7*n-4)*a(n-2) - 27*(n-2)*(n-1)*a(n-3) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 3^(2*n+9/2)/(16*Pi*n^3). - Vaclav Kotesovec, Oct 20 2012

a(n) = hypergeom([1/2, -n - 1, -n], [2, 2], 4). - Vaclav Kotesovec, May 14 2016

Illustrate a(n) = Sum_{k=0..n} A000108(k)*A001263(n+1,k+1) by:

a(2) = 1*(1) + 1*(3) + 2*(1) = 6;

a(3) = 1*(1) + 1*(6) + 2*(6) + 5*(1) = 24;

a(4) = 1*(1) + 1*(10)+ 2*(20)+ 5*(10)+ 14*(1) = 115.

where the Narayana triangle A001263(n+1,k+1) = C(n,k)*C(n+1,k)/(k+1) begins:

1;

1, 1;

1, 3, 1;

1, 6, 6, 1;

1, 10, 20, 10, 1;

1, 15, 50, 50, 15, 1; ...

a := n -> hypergeom([1/2, -n - 1, -n], [2, 2], 4):

seq(simplify(a(n)), n = 0..23); # Peter Luschny, Nov 06 2023

Table[Sum[Binomial[2*k, k]*Binomial[n, k]*Binomial[n+1, k]/(k+1)^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 20 2012 *)

Table[HypergeometricPFQ[{1/2, -1 - n, -n}, {2, 2}, 4], {n, 0, 20}] (* Vaclav Kotesovec, May 14 2016 *)

(PARI) {a(n)=sum(k=0, n, binomial(2*k, k)*binomial(n, k)*binomial(n+1, k)/(k+1)^2)}

Cf. A005802, A000108, A001263, A259833.

Sequence in context: A174072 A224255 A326348 * A069657 A369766 A211321

Adjacent sequences: A128085 A128086 A128087 * A128089 A128090 A128091

Paul D. Hanna, Feb 23 2007

approved