A203899 Number of support partitions-vertices.

1, 1, 2, 3, 3, 4, 5, 5, 8, 8, 8, 9, 13, 12, 15, 17, 20, 19, 28, 27, 42, 36, 42, 38, 53, 47, 70, 65, 79, 76, 100, 84, 119, 101, 140, 126, 169, 143, 189, 177, 233, 202, 291, 262, 367, 295, 391, 324, 460, 380, 523, 453, 599, 524, 716, 607, 818, 697, 914, 789

Offset

1,3

Comments

This sequence is the sequence of the numbers of support vertices of the integer partition polytopes. As in A203898, partitions of n are considered as the points x in R^n; the integer partition polytope P_n is the convex hull of all partitions of n. A vertex x of P_n is called support if it cannot be obtained from any other vertex of P_n with the use of any of the two operations of merging parts: (i) substituting x_u parts u of x by one part x_{u}u, (ii) substituting x_u parts u and x_v-x_u parts v of x, x_u <= x_v, by x_u parts u+v. These operations result in vertices if applied to vertices.

Support vertices of P_n form a basis for the set of partitions of n. This sequence was computed by A. S. Vroublevski.

References

Vladimir A. Shlyk, Combinatorial operations for generating vertices of integer partition polytopes, Dokl. Nats. Akad. Nauk Belarusi, 53/6 (2009), 27-32 (in Russian).

Links

Vladimir A. Shlyk, Table of n, a(n) for n = 1..100

Vladimir A. Shlyk, Polytopes of Partitions of Numbers, European J. Combin., Vol. 26/8 2005, 1139-1153.

Example

Application of the operation (i) with u=1 to the vertex x=(2,0,1,0,0) of P_5 results in the vertex y=(0,1,1,0,0), while application of the operation (ii) with u=3, v=1 to x results in the vertex z=(1,0,0,1,0). Hence, both y and z are not support vertices of P_5.

Crossrefs

Cf. A000041, A108917, A203898.

Sequence in context: A156562 A290736 A007998 * A202704 A273004 A029931

Adjacent sequences: A203896 A203897 A203898 * A203900 A203901 A203902

Keyword

nonn

Author

Vladimir A. Shlyk, Jan 07 2012

Extensions

Corrected, extended, and b-file added by Vladimir A. Shlyk, Apr 29 2012

Status

approved