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A364856 Triangle read by rows: T(n,k) is the number of k-dimensional faces of the n-dimensional Kunz cone, 0 <= k <= n.

%I A364856 #27 Sep 17 2023 11:03:06
%S A364856 1,1,1,1,2,1,1,4,4,1,1,8,14,8,1,1,11,29,30,12,1,1,30,114,152,84,18,1,
%T A364856 1,47,247,468,402,158,24,1,1,122,826,1934,2120,1166,306,32,1,1,225,
%U A364856 1981,6018,8703,6593,2616,504,40,1,1,812,8275,28480,47255,42650,21610,5980,830,50,1
%N A364856 Triangle read by rows: T(n,k) is the number of k-dimensional faces of the n-dimensional Kunz cone, 0 <= k <= n.
%C A364856 Beginning with the Kunz cone of dimension n=0, we list the number of faces in each dimension of that cone. Each cone has a single 0-dimensional vertex and a single n-dimensional face which is the entire cone, so each vector is bookended by 1s. The number of (n-1)-dimensional facets in the n-dimensional cone is A007590(n), and the number of ridges is given by a formula listed in O'Sullivan's thesis, see references. These values are computed; no formula exists for dimensions between and including 1 and n-3.
%D A364856 E. Kunz, Über die Klassifikation numerischer Halbgruppen, vol. 11 of Regensburger Mathematische Schriften [Regensburg Mathematical Publications], Universität Regensburg, Fachbereich Mathematik, Regensburg, 1987.
%D A364856 J. Rosales, P. García-Sánchez, Numerical semigroups, Developments in Mathematics, Vol. 20, Springer-Verlag, New York, 2009.
%H A364856 J. Autry, A. Ezell, T. Gomes, C. O'Neill, C. Preuss, T. Saluja, and E. Torres Davila, <a href="https://arxiv.org/abs/1912.04460">Numerical semigroups, polyhedra, and posets II: locating certain families of semigroups</a>, arXiv:1912.04460 [math.CO], 2019-2021; Adv. Geom., 22 (2022), pp. 22-48.
%H A364856 W. Bruns, P. García-Sánchez, C. O'Neill, and D. Wilburne, <a href="https://arxiv.org/abs/1903.04342">Wilf's conjecture in fixed multiplicity</a>, arXiv:1903.04342 [math.CO], 2019.
%H A364856 T. Gomes, C. O'Neill, and E. T. Davila, <a href="https://doi.org/10.37236/10380">Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension</a>, The Electronic Journal of Combinatorics, Volume 30, Issue 2 (2023) #P2.57.
%H A364856 N. Kaplan and C. O'Neill, <a href="https://doi.org/10.5070/C61055385">Numerical semigroups, polyhedra, and posets I: The group cone</a>, Comb. Theory, 1 (2021) pp. Paper No 19, 23.
%H A364856 Emily O'Sullivan, <a href="https://oeis.org/A007590/a007590.pdf">Understanding the face structure of the Kunz cone</a>, Master's thesis, San Diego State Univ., 2023.
%e A364856 We define the 0-dimensional cone to be a single point; thus there is 1 0-dimensional face (the vertex/the cone itself).
%e A364856 In the 1-dimensional cone there is 1 0-dimensional face (the vertex) and 1 1-dimensional face (the ray).
%e A364856 In the 2-dimensional cone there is 1 0-dimensional vertex, 2 1-dimensional rays, and 1 2-dimensional face.
%e A364856 In the 3-dimensional cone, there is 1 0-dimensional vertex, 4 1-dimensional rays, 4 2-dimensional facets, and 1 3-dimensional face.
%e A364856 Triangle begins:
%e A364856   1;
%e A364856   1,  1;
%e A364856   1,  2,   1;
%e A364856   1,  4,   4,   1;
%e A364856   1,  8,  14,   8,  1;
%e A364856   1, 11,  29,  30, 12,  1;
%e A364856   1, 30, 114, 152, 84, 18, 1;
%e A364856 ...
%Y A364856 For each n-dimensional cone, the number of (n-1)-dimensional facets is A007590(n) for n>=2.
%K A364856 nonn,tabl
%O A364856 0,5
%A A364856 _Emily O'Sullivan_, Aug 10 2023