A097877 Triangle read by rows: T(n,k) is the number of Dyck n-paths with k large components, 0 <= k <= n/2.

1, 1, 1, 1, 1, 4, 1, 12, 1, 1, 34, 7, 1, 98, 32, 1, 1, 294, 124, 10, 1, 919, 448, 61, 1, 1, 2974, 1576, 298, 13, 1, 9891, 5510, 1294, 99, 1, 1, 33604, 19322, 5260, 583, 16, 1, 116103, 68206, 20595, 2960, 146, 1, 1, 406614, 242602, 78954, 13704, 1006, 19, 1, 1440025

Offset: 0

Author

David Callan, Sep 21 2004

Keywords

• nonn
• tabf

Comments

A prime Dyck path is one with exactly one return to ground level. Every nonempty Dyck path decomposes uniquely as a concatenation of prime Dyck paths, called its components. For example, UUDDUD has 2 components: UUDD and UD, of semilength 2 and 1 respectively. A large component is one of semilength >= 2.

Conjecture: This is the statistic "indegree" of elements in Pallo's comb posets. For the statistic "outdegree", see A009766. - F. Chapoton, Apr 18 2023

Examples

			T(3,1) = 4 because each of the 5 Dyck paths of semilength 3 has one large component except for UDUDUD, which has none.
Table begins:
  \ k 0, 1, 2, ...
  n\ ____________________
  0 | 1;
  1 | 1;
  2 | 1,   1;
  3 | 1,   4;
  4 | 1,  12,   1;
  5 | 1,  34,   7;
  6 | 1,  98,  32,  1;
  7 | 1, 294, 124, 10;
  8 | 1, 919, 448, 61, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• J. M. Pallo, Right-arm rotation distance between binary trees, Inform. Process. Lett., 87(4):173-177, 2003.

Crossrefs

The Fine distribution (A065600) counts Dyck paths by number of small components (= number of low peaks).

Cf. A000108, A009766.

Formula

G.f.: 2/(1 + t*(1 - 4*z)^(1/2) + (1 - 2*z)(1-t)) = Sum_{n>=0, k>=0} T(n, k) z^n t^k satisfies (1-z)*G = 1 + z*t*(CatalanGF[z]-1)*G. The gf for Dyck paths (A000108) with z marking semilength is CatalanGF[z]:=(1 - sqrt[1 - 4*z])/(2*z). Hence the gf for prime Dyck paths is z*CatalanGF[z] and the gf for non-UD prime Dyck paths is S(z):= z*CatalanGF[z]-z. For fixed k, the gf for (T(n, k))_{n>=0} is S(z)^k/(1-z)^(k+1). This is clear because 1/(1-z) is the gf for all-UD Dyck paths (including the empty one) and a Dyck path with k large components is a product (uniquely) of an all-UD, a non-UD prime, an all-UD, a non-UD prime, ... with k non-UD primes and k+1 all-UDs.