A188919 Triangle read by rows: T(n,k) = number of permutations of length n with k inversions that avoid the "dashed pattern" 1-32.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 3, 3, 1, 1, 1, 2, 4, 7, 8, 9, 9, 6, 4, 1, 1, 1, 2, 4, 7, 13, 16, 22, 26, 29, 26, 23, 17, 10, 5, 1, 1, 1, 2, 4, 7, 13, 22, 31, 44, 60, 74, 89, 95, 98, 93, 82, 63, 47, 29, 15, 6, 1, 1, 1, 2, 4, 7, 13, 22, 38, 55, 83, 116, 160, 207, 259, 304, 347, 375, 386, 378, 348, 304, 249, 190, 131, 85, 46, 21, 7, 1

Offset

0,7

Comments

Row n has length 1 + binomial(n,2) and sum A000110(n) (a Bell number).

Links

Alois P. Heinz, Rows n = 0..50, flattened

A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.

Andrew Baxter, Additional terms, formatted as a table.

Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv preprint arXiv:1108.2642, 2011

Jean-Christophe Novelli, Jean-Yves Thibon, Frédéric Toumazet, Noncommutative Bell polynomials and the dual immaculate basis, arXiv:1705.08113 [math.CO], 2017.

Example

Triangle begins:
1
1
1 1
1 1 2 1
1 1 2 4 3 3 1
1 1 2 4 7 8 9 9 6 4 1
...

Maple

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(u-j, o+j-1)*x^(o+j-1), j=1..u)+
       add(`if`(u=0, b(u+j-1, o-j)*x^(o-j), 0), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(0, n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 14 2015

Mathematica

b[u_, o_] := b[u, o] = Expand[If[u+o == 0, 1, Sum[b[u-j, o+j-1]* x^(o+j-1), {j, 1, u}] + Sum[If[u == 0, b[u+j-1, o-j]*x^(o-j), 0], {j, 1, o}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}] ][b[0, n]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 01 2016, after Alois P. Heinz *)

Crossrefs

The column limits are given by A188920.

Cf. A000110, A161680.

Sequence in context: A267383 A332648 A272896 * A026519 A025177 A026148

Adjacent sequences: A188916 A188917 A188918 * A188920 A188921 A188922

Keyword

nonn,tabf

Author

N. J. A. Sloane, Apr 13 2011

Extensions

More terms from Andrew Baxter, May 17 2011.

Status

approved