A236406 Triangle read by rows: number of (1-2-3)-avoiding permutations on n letters with k peaks.

1, 1, 2, 3, 2, 4, 10, 5, 32, 5, 6, 84, 42, 7, 198, 210, 14, 8, 438, 816, 168, 9, 932, 2727, 1152, 42, 10, 1936, 8250, 5940, 660, 11, 3962, 23276, 25630, 5775, 132, 12, 8034, 62400, 97812, 37180, 2574, 13, 16200, 160953, 341224, 196625, 27456, 429, 14, 32556, 402906, 1111656, 905086, 212212, 10010

Offset: 0

Author

N. J. A. Sloane, Jan 31 2014

Keywords

• nonn
• tabf

Comments

This is a convolution of A091156 with itself (see the Pudwell link below).

Examples

			Triangle begins:
   1;
   1;
   2;
   3,    2;
   4,   10;
   5,   32,    5;
   6,   84,   42;
   7,  198,  210,   14;
   8,  438,  816,  168;
   9,  932, 2727, 1152,  42;
  10, 1936, 8250, 5940, 660;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..120, flattened
• A. M. Baxter, Refining enumeration schemes to count according to permutation statistics, arXiv preprint arXiv:1401.0337 [math.CO], 2014.
• M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
• L. Pudwell, On the distribution of peaks (and other statistics), 2018.

Crossrefs

Row sums give A000108.

Cf. A091894, A276666.

Programs

Mathematica

m = maxExponent = 15;
G = -(-2 z^3 q^2 + 4z^3 q - 2z^3 - 2z^2 q + 2z^2 - 1 + Sqrt[-4z^2 q + 4z^2 - 4z + 1])/(2z (z q - z + 1)^2);
CoefficientList[# + O[q]^m, q]& /@ CoefficientList[G + O[z]^m, z]// Flatten (* Jean-François Alcover, Aug 06 2018 *)

Formula

T(2*n+2,n) = A276666(n+2) = (n+1)*A000108(n+2). - Alois P. Heinz, Apr 27 2018

G.f.: G(q,z) = - (-2z^3q^2+4z^3q-2z^3-2z^2q+2z^2-1+sqrt(-4z^2q+4z^2-4z+1))/(2z(zq-z+1)^2). (See the Pudwell link above.)

Extensions

More terms from Alois P. Heinz, Apr 26 2018