OFFSET
1,2
COMMENTS
This sequence is the sequence of the numbers of vertices of the integer partition polytopes. Partitions of n are considered as the points x in R^n: each x_i is the number of times the part i enters the partition. The partition polytope P_n is the convex hull of all partitions of n.
This sequence is dominated by A108917 since each vertex is proved to be a knapsack partition. This sequence was computed by A. S. Vroublevski with the sporadic aid of Polymake.
REFERENCES
Vladimir A. Shlyk, Polytopes of partitions of numbers, Vesti Ac. Sci. Belarus, Ser. phys.-mat. nauk, No. 3 (1996), 89-92 (in Russian).
Vladimir A. Shlyk, On the vertices of the polytopes of partitions of numbers, Dokl. Nats. Akad. Nauk Belarusi, 52.3 (2008), 5-10 (in Russian).
LINKS
Vladimir A. Shlyk, Table of n, a(n) for n = 1..105
Shmuel Onn and Vladimir A. Shlyk, Some Efficiently Solvable Problems over Integer Partition Polytopes, Discrete Appl. Math., Vol. 180 2015, 135-140.
Vladimir A. Shlyk, Polytopes of Partitions of Numbers, European J. Combin., Vol. 26/8 2005, 1139-1153.
Vladimir A. Shlyk, Polyhedral Approach to Integer Partitions, J. Combin. Math. Combin. Computing, Vol. 89 2014, 113-128.
Vladimir A. Shlyk, Recursive Operations for Generating Vertices of Integer Partition Polytopes, Communication of the Joint Institute for Nuclear Research, E5-2008-18. Dubna, 2008.
Vladimir A. Shlyk, A Criterion of Representability of an Integer Partition as a Convex Combination of Two Partitions, Vestnik BGU. Ser. 1, No. 2 (2009), 109-114 (in Russian).
Vladimir A. Shlyk, Combinatorial Operations for Generating Vertices of Integer Partition Polytopes, Dokl. Nats. Akad. Nauk Belarusi, 53.6 (2009), 27-32 (in Russian).
Vladimir A. Shlyk, On the Relation of Vertices of Integer Partition Polytopes to Their Nontrivial Facets, Vestnik BGU. Ser. 1, No. 1 (2010), 153-156 (in Russian).
Vladimir A. Shlyk, On the Adjacency of Vertices of the Integer Partition Polytope. Part I, Izvestia National Acad. Sci. Ser. Phys.-Math. Sci, No. 1 (2011), 112-117 (in Russian).
Vladimir A. Shlyk, On the Adjacency of Vertices of the Integer Partition Polytope. Part II, Izvestia National Acad. Sci. Ser. Phys.-Math. Sci, No. 3 (2011), 105-111 (in Russian).
Vladimir A. Shlyk, Integer Partitions from the Polyhedral Point of View, Electron. Notes Discrete Math, Vol. 43 2013, 319-327.
Vladimir A. Shlyk, Master Corner Polyhedron: Vertices, Eur. J. Oper. Res., 226/2 (2013), 203-210.
Vladimir A. Shlyk, Number of Vertices of the Polytope of Integer Partitions and Factorization of the Partitioned Number, arXiv:1805.07989 [math.CO], 2018.
A. S. Vroublevski and Vladimir A. Shlyk, Computing vertices of integer partition polytopes, Computing vertices of integer partition polytopes, Informatics, 48.4 (2015), 34-48 (in Russian).
EXAMPLE
The partition x=(2,1,0,0) of 4 corresponds to 4=2+1+1. It is not a vertex of P_4 since x=((4,0,0,0)+(0,2,0,0))/2. The partition x=(0,0,2,1,1,0^{10}) of n=15 is the first partition that is a convex combination of 3 partitions: x=((0,0,0,0,3,0^{10})+(0,0,1,3,0,0^{10})+(0,0,5,0,0,0^{10})/3.
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir A. Shlyk, Jan 07 2012
STATUS
approved