A Combinatorial Proof of a Relationship Between Maximal $(2k-1, 2k+1)$-Cores and $(2k-1, 2k, 2k+1)$-Cores

Retracts of Odd-angulated Graphs and Construction of Cores

Let G be a connected graph with odd girth 2k + 1. Then G is a (2k + 1)-angulated graph if every two vertices of G are joined by a path such that each edge of the path is in some (2k + 1)-cycle. We prove that if G is (2k+1)-angulated, and H is connected with odd girth at least 2k+3, then any retract R of the box (or Cartesian) product G2H is isomorphic to S2T where S is a retract of G and T is a...

PARITY RESULTS FOR BROKEN k–DIAMOND PARTITIONS AND (2k + 1)–CORES

In this paper we prove several new parity results for broken k-diamond partitions introduced in 2007 by Andrews and Paule. In the process, we also prove numerous congruence properties for (2k+1)-core partitions. The proof technique involves a general lemma on congruences which is based on modular forms.

Appendix for FCD Load Balancing under Switching Costs and Imperfect Observations

I. PROOF OF PROPOSITION 1 Proof: Consider a simple example of n = 2 servers in a super time slot. At t = 2k, assume without loss of generality that X 1 (2k) > X 2 (2k). At the exploration slot t = 2k+1 with exploration probability γ 2k , the expected number of users is thus E [X 1 (2k + 1)] = X 1 (2k)(1 − γ 2k) + X 2 (2k)γ 2k E [X 2 (2k + 1)] = X 2 (2k)(1 − γ 2k) + X 1 (2k)γ 2k. On one hand, ea...

A minimum-change version of the Chung-Feller theorem for Dyck paths

A Dyck path with 2k steps and e flaws is a path in the integer lattice that starts at the origin and consists of k many ↗-steps and k many ↘-steps that change the current coordinate by (1, 1) or (1,−1), respectively, and that has exactly e many ↘-steps below the line y = 0. Denoting by D 2k the set of Dyck paths with 2k steps and e flaws, the Chung-Feller theorem asserts that the sets D 2k, D 1...

Some Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points