A335349 - OEIS

A335349

a(n) counts anti-chains of size three in "0,1,2" Motzkin trees on n edges.

2, 16, 98, 500, 2308, 9920, 40522, 159212, 606790, 2256544, 8224202, 29473012, 104124044, 363374560, 1254711038, 4292365876, 14564351510, 49059814576, 164186524940, 546276316120, 1807990549352, 5955265349696, 19530431537488, 63795464433440, 207623760855106, 673440401953856

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OFFSET

4,1

COMMENTS

"0,1,2" trees are rooted trees where each vertex has outdegree zero, one, or two. They are counted by the Motzkin numbers A001006.

A005717(n+1) is the total number of vertices (= anti-chains of size 1) in all "0,1,2" trees with n edges, while A178834(n) is the total number of anti-chains of size 2 in all "0,1,2" trees on n edges.

LINKS

Table of n, a(n) for n=4..29.

Lifoma Salaam, Combinatorial statistics on phylogenetic trees, Ph.D. Dissertation, Howard University, Washington D.C., 2008; see Definition 42 (p. 30), Theorem 44 (p. 33), and Table 2.4 (p. 39).

FORMULA

G.f. is A000108(r-1) * z^(2*r-2) * T(z)^(2*r-1) * M(z)^r = 2 * z^4 * T(z)^5 * M(z)^3 (with r = 3), where M(z) = (1 - z - sqrt(1 - 2*z - 3*z^2)) / (2*z^2) is the g.f. of the Motzkin numbers A001006 and T(z) = 1 / sqrt(1 - 2*z - 3*z^2) is the g.f. of the central trinomial numbers A002426.

EXAMPLE

Out of the A001006(4) = 9 Motzkin rooted trees, there are only two that have anti-chains of size 3 (i.e., 3-sets of pairwise incomparable nodes), and each one has only one such an anti-chain. Thus, a(4) = 1 + 1 = 2.

In the first Motzkin tree below with 4 edges, {E, C, D} is an anti-chain of size 3. In the second one, {G, I, K} is an anti-chain of size 3.

A F

/ \ / \

B E G H

/ \ / \

C D I K

PROG

(PARI) M(z) = (1 - z - sqrt(1 - 2*z - 3*z^2))/(2*z^2);

T(z) = 1/sqrt(1 - 2*z - 3*z^2);

my(z='z+O('z^30)); Vec(2*z^4*T(z)^5*M(z)^3)

CROSSREFS

Cf. A000108, A001006, A002426, A005717, A178834.