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T(n, k) = number of k-dimensional faces in the BME polytope on n species, 0 <= k <= binomial(n, 2) - n.
1

%I #19 Jul 29 2023 02:59:38

%S 3,3,1,15,105,250,210,52,1,105,5460,105945,635265,1715455,2373345,

%T 1742445,640140,90262,1

%N T(n, k) = number of k-dimensional faces in the BME polytope on n species, 0 <= k <= binomial(n, 2) - n.

%C The balanced minimum evolution (BME) polytope of order n is the convex hull of the BME vectors of all phylogenetic trees on n species. The BME polytope of order n has dimension binomial(n, 2) - n.

%H Maria Angelica Cueto and Frederick A. Matsen, <a href="https://doi.org/10.1007/s11538-010-9556-x">Polyhedral geometry of phylogenetic rogue taxa</a>, Bull. Math. Biol., 73 (2011), 1202-1226.

%H K. Eickmeyer, P. Huggins, L. Pachter, and R. Yoshida, <a href="https://doi.org/10.1186%2F1748-7188-3-5">On the optimality of the neighbor-joining algorithm</a>, Algorithms Mol Biol. 3 (2008), Article number 5.

%H Stefan Forcey, <a href="https://sforcey.github.io/sf34/hedra.htm#BME">Balanced Minimum Evolution Polytope</a>, Encyclopedia of Combinatorial Polytope Sequences (Hedra Zoo).

%e Table begins:

%e 3, 3, 1;

%e 15, 105, 250, 210, 52, 1;

%e 105, 5460, 105945, 635265, 1715455, 2373345, 1742445, 640140, 90262, 1;

%Y First column T(n, 0) is A001147.

%Y Next-to-last entry T(n, binomial(n, 2) - n - 1) in each row is A364441.

%K nonn,tabf,more

%O 4,1

%A _Harry Richman_, Jul 26 2023