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A306350
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Number of paraphyletic coalescence sequences for 2n lineages, n each in 2 species.
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1
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0, 4, 162, 23328, 9072000, 7873200000, 13367512620000, 40367907740160000, 201793403949096960000, 1578804075215377920000000, 18484433452834116768000000000, 312162837144268369009766400000000, 7374810540967959718955457331200000000
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OFFSET
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1,2
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COMMENTS
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Consider a binary tree evolving in time from a single node until the tree has 2n labeled leaves. Color the 2n leaves in 2 colors, red and blue, assigning n leaves to each color. Suppose coalescences of pairs of leaves happen at distinct times (i.e., no simultaneous mergers). A coalescence sequence is a sequence of coalescence events backward in time, tracing the reduction of the 2n leaves to the single ancestral node. A paraphyletic coalescence sequence is a sequence in which (1) all n red leaves have a common ancestor node that is not the ancestor of any blue leaves; or (2) all n blue leaves have a common ancestor node that is not the ancestor of any red leaves; but not both (1) and (2).
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LINKS
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FORMULA
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a(n) = 3 n! n! (2n-2)! (n-1) / ( (n+1) 2^(2n-3) ).
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EXAMPLE
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For n=2, consider two red leaves R1 and R2 and two blue leaves B1 and B2. The a(2)=4 paraphyletic coalescence sequences, separated by semicolons, are (R1,R2), ((R1,R2),B1), (((R1,R2),B1),B2); (R1,R2), ((R1,R2),B2), (((R1,R2),B2),B1); (B1,B2), ((B1,B2),R1), (((B1,B2),R1),R2); and (B1,B2), ((B1,B2),R2), (((B1,B2),R2),R1).
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MATHEMATICA
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Table[3 n! n! (2 n - 2)! (n - 1)/((n + 1) (2^(2 n - 3))), {n, 1, 30}]
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CROSSREFS
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The total number of coalescence sequences for n leaves, from among which the paraphyletic coalescence sequences are identified, follows A006472. Reciprocally monophyletic coalescence sequences, in which conditions (1) and (2) above both hold, follow A306266.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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