|
%I #28 Mar 28 2023 17:12:28
%S 1,1,7,106,2575,87595,3864040,210455470,13681123135,1035588754375,
%T 89575852312675,8724157965777400,945424197750836500,
%U 112891958206958894500,14733016566584898017500,2086947723639167040631750,318968341048949169038143375
%N 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.
%C Also the number of tree-child networks with a maximal number n of reticulations nodes. - _Michael Fuchs_, Aug 05 2020
%H Alois P. Heinz, <a href="/A213863/b213863.txt">Table of n, a(n) for n = 0..320</a>
%H Cyril Banderier and Michael Wallner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/jenga2021.pdf">Young tableaux with periodic walls: counting with the density method</a>, Séminaire Lotharingien de Combinatoire XX, Proceedings of the 33rd Conference on Formal Power (2021) Article #YY.
%H Michael Fuchs, <a href="https://web.math.nccu.edu.tw/mfuchs/TCNs-talk.pdf">Enumeration and Stochastic Properties of Tree-Child Networks</a>, National Chengchi Univ. (Taipei 2023).
%H Michael Fuchs, Guan-Ru Yu, and Louxin Zhang, <a href="https://arxiv.org/abs/2003.08049">On the Asymptotic Growth of the Number of Tree-Child Networks</a>, arXiv:2003.08049 [math.CO], 2020.
%F 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
%e a(0) = 1: the empty word.
%e a(1) = 1: aaa.
%e a(2) = 7: aaabbb, aababb, aabbab, abaabb, ababab, baaabb, baabab.
%Y Row n=3 of A213275.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jun 23 2012
|
