A265164 Sum of the n-th row of the array A265163(n, k).

1, 3, 15, 101, 841, 8283, 93815, 1198029, 16997041, 264864419, 4492081151, 82299283669, 1618674299769, 33997164987019, 759059595497511, 17945237236457533, 447676430154815137, 11748882878147100691, 323494584038834863087, 9322205037165367256837

Offset: 0

Author

Cyril Banderier, Dec 07 2015; revised Feb 06 2017

Keywords

• nonn

Comments

A right-jump in a permutation consists of taking an element and moving it somewhere to its right.

The set P(k) of permutations reachable from the identity after at most k right-jumps is a permutation-pattern avoiding set: it coincides with the set of permutation avoiding a set of patterns.

We define B(k) to be the smallest such set of "forbidden patterns" (the permutation pattern community calls such a set a "basis" for P(k), and its elements can be referred to as "right-jump basis permutations").

The number b(n,k) of permutations of size n in B(k) is given by the array A265163.

The row sums give the present sequence (i.e. this counts the permutations of any size in the basis B(k)).

The column sums give the sequence A265165 (i.e. this counts the permutations of size n in any B(k)).

Examples

			G.f. = 1 + 3*x + 15*x^2 + 101*x^3 + 841*x^4 + 8283*x^5 + 93815*x^6 + 1198029*x^7 + ...
The basis permutations for B(1) are 312, 321, and 2143, thus a(1)=3.
The basis permutations for B(2) are 4123, 4132, 4213, 4231, 4312, 4321, 21534, 21543, 31254, 32154, 31524, 31542, 32514, 32541, and 214365, thus a(2)=15.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200
• Cyril Banderier, Jean-Luc Baril, Céline Moreira Dos Santos, Right jumps in permutations, Permutation Patterns 2015.

Crossrefs

Cf. A265163, A265165.

Programs

Mathematica

a[ n_] := Module[ {A, s, F}, If[ n < 0, 0, A = 1 - x + O[x]^(2 n + 3); s = Sqrt[1 + 4 y + O[y]^(n + 2)]; F = y ((1 - 1/s) A^((1 + s)/2) + (1 + 1/s) A^((1 - s)/2))/2; Sum[ SeriesCoefficient[ SeriesCoefficient[ F, {x, 0, n + k}] (n + k)!, {y, 0, k}], {k, 2, 2 + n}]]]; (* Michael Somos, Jan 27 2017 *)

PARI

{a(n) = my(A, s, F); if( n<0, 0, A = 1 - x + x * O(x^(2*n+2)); s = sqrt(1 + 4*y + y * O(y^(n+1))); F = y * ((1 - 1/s) * A^((1 + s)/2) + (1 + 1/s) * A^((1 - s)/2)) / 2; sum(k=2, 2+n, polcoeff( polcoeff( F, n+k) * (n+k)!, k)))}; /* Michael Somos, Jan 27 2017 */