On the number of nonnegative sums for semi-partitions

Document Type

Article

Publication Date

11-1-2021

Abstract

Let n] = {1, 2,..., n}. Let (n] k) be the family of all subsets of n] of size k. Define a real-valued weight function w on the set n] k such that Sigma(X is an element of)(n] k) w( X) >= 0. Let U-n,U- t,U-k be the set of all P = {P-1, P-2,..., P-t} such that P-i. is an element of (n] k) for all i and P-i n P-j = O for i not equal j. For each P is an element of U-n,U- t,U-k, let w(P) = P is an element of P w( P). Let U+ n,t,k(w) be set of all P is an element of U-n,U- t,U-k with w(P) >= 0. In this paper, we showthat vertical bar U-n,t,k(+) (w)vertical bar >=Pi(1=i=(t-1)k) (n-tk+i)/(k!)t-1((t-1)!) for sufficiently large n.

Keywords

Subset sums, Extremal problems, Partitions

Publication Title

Graphs and Combinatorics

Divisions

MathematicalSciences

Volume

37

Issue

6

Publisher

Springer Verlag

Publisher Location

SHIROYAMA TRUST TOWER 5F, 4-3-1 TORANOMON, MINATO-KU, TOKYO, 105-6005, JAPAN

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