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A203899
Number of support partitions-vertices.
2
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, 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
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