A091866 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having pyramid weight k.

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 1, 5, 8, 0, 0, 1, 7, 18, 16, 0, 0, 1, 9, 34, 56, 32, 0, 0, 1, 11, 55, 138, 160, 64, 0, 0, 1, 13, 81, 275, 500, 432, 128, 0, 0, 1, 15, 112, 481, 1205, 1672, 1120, 256, 0, 0, 1, 17, 148, 770, 2471, 4797, 5264, 2816, 512, 0, 0, 1, 19, 189, 1156, 4536, 11403, 17738, 15808, 6912, 1024

Offset: 0

Author

Emeric Deutsch, Mar 10 2004

Keywords

• nonn
• tabl

Comments

A pyramid in a Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A pyramid in a Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a u and immediately followed by a d. The pyramid weight of a Dyck path (word) is the sum of the heights of its maximal pyramids.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 0, 1, 0, 0, 1, 0, 0, 1, ...] (periodic sequence 0,0,1) DELTA [1, 1, 0, 1, 1, 0, 1, 1, 0, ...] (periodic sequence 1,1,0), where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 18 2006

Peter Luschny observes that one of the rows of this triangle seems to appear on page 26 of Knuth (2014). - N. J. A. Sloane, Aug 02 2014

Examples

			T(4,3)=5 because the Dyck paths of semilength 4 having pyramid weight 3 are: (ud)u(ud)(ud)d, u(ud)(ud)d(ud), u(ud)(ud)(ud)d, u(ud)(uudd)d and u(uudd)(ud)d [here u=(1,1), d=(1,-1) and the maximal pyramids, of total length 3, are shown between parentheses].
Triangle begins:
  1;
  0,   1;
  0,   0,   2;
  0,   0,   1,   4;
  0,   0,   1,   5,   8;
  0,   0,   1,   7,  18,  16;
  0,   0,   1,   9,  34,  56,  32;
  0,   0,   1,  11,  55, 138, 160,  64;
  0,   0,   1,  13,  81, 275, 500, 432, 128;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Xiaomei Chen, Yuan Xiang, Counting generalized Schröder paths, arXiv:2009.04900 [math.CO], 2020.
• A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
• D. E. Knuth, Problems That Philippe Would Have Loved, Paris 2014.

Programs

Mathematica

nmax=11;
DELTA[r_, s_] := Module[{m=Min[Length[r], Length[s]], p, q, t, x, y}, q[k_] := x*r[[k+1]] + y*s[[k+1]]; p[0, ] = 1; p[, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k-1] + q[k]*p[n-1, k+1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n-k)*y^k]; t[0, 0] = p[0, 0]; Table[ t[n, k], {n, 0, m}, {k, 0, n}]];
Table[Mod[1+2n^2,3], {n,nmax}] ~DELTA~ Table[1-Mod[1+2n^2,3], {n,nmax}] (* Jean-François Alcover, Jun 06 2019 *)

Formula

G.f.: G = G(t, z) satisfies z(1-tz)G^2-(1+z-2tz)G+1-tz = 0.

Sum_{k=0..n} T(n,k) = A000108(n). - Philippe Deléham, Aug 18 2006