A213863 Number of words w where each letter of the n-ary alphabet occurs 3 times and for every prefix z of w we have #(z,a_i) = 0 or #(z,a_i) >= #(z,a_j) for all j>i and #(z,a_i) counts the occurrences of the i-th letter in z.

1, 1, 7, 106, 2575, 87595, 3864040, 210455470, 13681123135, 1035588754375, 89575852312675, 8724157965777400, 945424197750836500, 112891958206958894500, 14733016566584898017500, 2086947723639167040631750, 318968341048949169038143375

Offset

0,3

Comments

Also the number of tree-child networks with a maximal number n of reticulations nodes. - Michael Fuchs, Aug 05 2020

Links

Alois P. Heinz, Table of n, a(n) for n = 0..320

Cyril Banderier and Michael Wallner, Young tableaux with periodic walls: counting with the density method, Séminaire Lotharingien de Combinatoire XX, Proceedings of the 33rd Conference on Formal Power (2021) Article #YY.

Michael Fuchs, Enumeration and Stochastic Properties of Tree-Child Networks, National Chengchi Univ. (Taipei 2023).

Michael Fuchs, Guan-Ru Yu, and Louxin Zhang, On the Asymptotic Growth of the Number of Tree-Child Networks, arXiv:2003.08049 [math.CO], 2020.

Formula

a(n) = Sum_{m>=1} b_{n,m} if n>0. Here, b_{n,m} satisfies b_{n,m}=(2*n+m-2)*Sum_{k=1..m} b_{n-1,k} for n>=2 and 1<=m<=n with initial conditions b_{n,m}=0 for n<m and b_{1,1}=1. - Michael Fuchs, Aug 05 2020

Example

a(0) = 1: the empty word.
a(1) = 1: aaa.
a(2) = 7: aaabbb, aababb, aabbab, abaabb, ababab, baaabb, baabab.

Crossrefs

Row n=3 of A213275.

Sequence in context: A367166 A141358 A141362 * A231899 A188407 A075021

Adjacent sequences: A213860 A213861 A213862 * A213864 A213865 A213866

Keyword

nonn

Author

Alois P. Heinz, Jun 23 2012

Status

approved