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%I #30 Mar 06 2022 08:42:21
%S 0,1,12,151,2545,54466,1417318,43472780,1536228588,61466251616,
%T 2746907348768,135619260805568,7331022129923648,430638151053316480,
%U 27315015477709844352,1860627613021322933248,135465573609158928964096,10498038569346091127451136,862792664850194915870874112
%N Numbers of labeled rooted Greg trees (A005264) with n nodes and root degree 2.
%C Numbers of rooted Greg trees with 2 subtrees below root given m labeled nodes (lead index). Among all trees at the same index (see sequence A005264) root bifurcating trees play a central role in philological discourse on the reconstruction of manuscript genealogies. Labeled nodes represent surviving manuscripts, unlabeled nodes hypothetical ones. See also stemmatology/stemmatics, Bédier's paradox.
%D J. Bédier. La tradition manuscrite du Lai de l'Ombre: Réflexions sur l'Art d' Éditer les Anciens Textes. Romania 394 (1928), 161-196/321-356.
%D C. Flight. How many stemmata? Manuscripta 34(2), 1990, 122-128.
%D W. Hering. Zweispaltige Stemmata. Philologus-Zeitschrift für antike Literatur und ihre Rezeption 111(1-2), (1967), 170-185.
%D P. Maas. Textkritik. 4. Auflage. Leipzig: Teubner. 1960.
%H Armin Hoenen, <a href="/A286432/b286432.txt">Table of n, a(n) for n = 1..245</a>
%H Armin Hoenen, S. Eger and R. Gehrke, <a href="http://dx.doi.org/10.18653/v1/W17-3402">How many stemmata with root degree k?</a>, Proceedings of MOL 2017, 2017.
%F T_{m,2} = Sum_{n >= 0} T_{m,n,2}, where T_{m,n,k} = (m/k!) * Sum_{(s,p) in C((m-1,n),k)} (binomial(m-1,s) F(s,p)) + (1/k!) * Sum_{(s,p) in C((m,n-1),k)} (binomial(m,s) F(s,p)), with F(s,p) = Product_{1..k} (g(s_i,p_i)), here g(m,n) = numbers of rooted Greg trees, see (A005264) with m labeled and n unlabeled nodes. s and p are tuples with k elements where each s_i >= 1 and for each p_i : 0 <= p_i < s_i; first term in T_{m,n,k} gives the number of trees with a labeled root, second those for root unlabeled.
%e For n=3, T_{3,2} is T_{3,0,2} + T_{3,1,2} + T_{3,2,2} where T_{3,0,2} = (3/2) * (binomial(2,(1,1)) * product(g(1,0)*g(1,0))) + 0 = 3; T_{3,1,2} = 0 + 1/2 * ((binomial(3,(2,1)) * product(g(2,0)*g(1,0))) + (binomial(3,(1,2)) * product(g(1,0)*g(2,0)))) = 6 and T_{3,2,2} = 0 + (1/2) * ((binomial(3,(2,1)) * product(g(2,1)*g(1,0))) + (binomial(3,(1,2)) * product(g(1,0)*g(2,1)))) = 3; 3 + 6 + 3 =12.
%Y Cf. A005264, number of labeled rooted Greg trees with n nodes.
%Y Cf. A005263, unrooted Greg trees, according to Flight (1990) can also serve as basis for computation of A005624.
%K nonn
%O 1,3
%A _Armin Hoenen_, May 09 2017
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