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

0,6

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

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.

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

Example

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;
  ...

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 *)

Crossrefs

Sequence in context: A073429 A123634 A330140 * A168511 A111146 A109077

Adjacent sequences: A091863 A091864 A091865 * A091867 A091868 A091869

Keyword

nonn,tabl

Author

Emeric Deutsch, Mar 10 2004

Status

approved