A236406 - OEIS

%I #43 Dec 19 2018 11:23:10

%S 1,1,2,3,2,4,10,5,32,5,6,84,42,7,198,210,14,8,438,816,168,9,932,2727,

%T 1152,42,10,1936,8250,5940,660,11,3962,23276,25630,5775,132,12,8034,

%U 62400,97812,37180,2574,13,16200,160953,341224,196625,27456,429,14,32556,402906,1111656,905086,212212,10010

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

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

%H Alois P. Heinz, <a href="/A236406/b236406.txt">Rows n = 0..120, flattened</a>

%H A. M. Baxter, <a href="http://arxiv.org/abs/1401.0337">Refining enumeration schemes to count according to permutation statistics</a>, arXiv preprint arXiv:1401.0337 [math.CO], 2014.

%H M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, <a href="https://arxiv.org/abs/1812.07112">Distributions of Statistics over Pattern-Avoiding Permutations</a>, arXiv preprint arXiv:1812.07112 [math.CO], 2018.

%H L. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/PP2018Pudwell.pdf">On the distribution of peaks (and other statistics)</a>, 2018.

%F T(2*n+2,n) = A276666(n+2) = (n+1)*A000108(n+2). - _Alois P. Heinz_, Apr 27 2018

%F 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.)

%e Triangle begins:

%e    1;

%e    1;

%e    2;

%e    3,    2;

%e    4,   10;

%e    5,   32,    5;

%e    6,   84,   42;

%e    7,  198,  210,   14;

%e    8,  438,  816,  168;

%e    9,  932, 2727, 1152,  42;

%e   10, 1936, 8250, 5940, 660;

%e   ...

%t m = maxExponent = 15;

%t 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);

%t CoefficientList[# + O[q]^m, q]& /@ CoefficientList[G + O[z]^m, z]// Flatten (* _Jean-François Alcover_, Aug 06 2018 *)

%Y Row sums give A000108.

%Y Cf. A091894, A276666.

%K nonn,tabf

%O 0,3

%A _N. J. A. Sloane_, Jan 31 2014

%E More terms from _Alois P. Heinz_, Apr 26 2018