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%I #8 Oct 18 2020 13:59:35
%S 1,2,7,40,358,4884,99665,3000736,131932016,8403206128
%N Number of vertices of the Gelfand Tsetlin-polytope. Alternatively, the number of Gelfand-Tsetlin patterns with top row 1234...n and such that every entry in a given row also appears in the row above it.
%C It is easy to mistake these for monotone triangles.
%H Luca De Feo, David Jao and Jerome Plut, <a href="https://eprint.iacr.org/2011/506">Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies</a>, Cryptology ePrint Archive: Report 2011/506. - From _N. J. A. Sloane_, Dec 22 2012
%e a(3)=7 because the vertices of GT(3) are
%e 123
%e 12
%e 1
%e ---
%e 123
%e 12
%e 2
%e ---
%e 123
%e 13
%e 1
%e ---
%e 123
%e 13
%e 3
%e ---
%e 123
%e 23
%e 2
%e ---
%e 123
%e 23
%e 3
%e ---
%e 123
%e 22
%e 2
%e ---
%t (* G computes the required sequence, F computes the similar sequence with any monotone sequence permitted as the input top row. Note that F and Bifurcate cache their values. *) Bifurcate[l_] := Bifurcate[l] = If[Length[l] == 1, { {} }, Union[Map[Prepend[ #, l[[1]]] &, Bifurcate[Drop[l, 1]]], Map[ Prepend[ #, l[[2]]] &, Bifurcate[Drop[l, 1]]]]] F[l_] := F[l] = If[Length[l] == 0, 1, Apply[Plus, Map[F, Bifurcate[l]]]] G[n_] := F[Range[n]]
%K nonn,more
%O 1,2
%A David E Speyer (speyer(AT)post.harvard.edu), Jul 02 2007
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