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

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, 1, 47, 247, 468, 402, 158, 24, 1, 1, 122, 826, 1934, 2120, 1166, 306, 32, 1, 1, 225, 1981, 6018, 8703, 6593, 2616, 504, 40, 1, 1, 812, 8275, 28480, 47255, 42650, 21610, 5980, 830, 50, 1

Offset

0,5

Comments

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.

References

E. Kunz, Über die Klassifikation numerischer Halbgruppen, vol. 11 of Regensburger Mathematische Schriften [Regensburg Mathematical Publications], Universität Regensburg, Fachbereich Mathematik, Regensburg, 1987.

J. Rosales, P. García-Sánchez, Numerical semigroups, Developments in Mathematics, Vol. 20, Springer-Verlag, New York, 2009.

Links

Table of n, a(n) for n=0..65.

J. Autry, A. Ezell, T. Gomes, C. O'Neill, C. Preuss, T. Saluja, and E. Torres Davila, Numerical semigroups, polyhedra, and posets II: locating certain families of semigroups, arXiv:1912.04460 [math.CO], 2019-2021; Adv. Geom., 22 (2022), pp. 22-48.

W. Bruns, P. García-Sánchez, C. O'Neill, and D. Wilburne, Wilf's conjecture in fixed multiplicity, arXiv:1903.04342 [math.CO], 2019.

T. Gomes, C. O'Neill, and E. T. Davila, Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension, The Electronic Journal of Combinatorics, Volume 30, Issue 2 (2023) #P2.57.

N. Kaplan and C. O'Neill, Numerical semigroups, polyhedra, and posets I: The group cone, Comb. Theory, 1 (2021) pp. Paper No 19, 23.

Emily O'Sullivan, Understanding the face structure of the Kunz cone, Master's thesis, San Diego State Univ., 2023.

Example

We define the 0-dimensional cone to be a single point; thus there is 1 0-dimensional face (the vertex/the cone itself).
In the 1-dimensional cone there is 1 0-dimensional face (the vertex) and 1 1-dimensional face (the ray).
In the 2-dimensional cone there is 1 0-dimensional vertex, 2 1-dimensional rays, and 1 2-dimensional face.
In the 3-dimensional cone, there is 1 0-dimensional vertex, 4 1-dimensional rays, 4 2-dimensional facets, and 1 3-dimensional face.
Triangle begins:
  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;
...

Crossrefs

For each n-dimensional cone, the number of (n-1)-dimensional facets is A007590(n) for n>=2.

Sequence in context: A064298 A256894 A372068 * A099594 A255256 A328887

Adjacent sequences: A364853 A364854 A364855 * A364857 A364858 A364859

Keyword

nonn,tabl

Author

Emily O'Sullivan, Aug 10 2023

Status

approved