2025/06/23 by Abdurakhmanov, Jamolidin K.
#Discrete Mathematics (cs.DM) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2506.18494
We present a novel framework for studying combinatorial identities through the geometric lens of subset distributions in q-valued cubes. By analyzing how elements of arbitrary subsets are distributed among the faces of the cube Eqn, we discover new combinatorial identities with geometric significance. We prove that for any subset A contained in E2n, the rank function satisfies refined bounds that lead to exact computations for small cardinalities. Specifically, we show that for odd cardinalities, the lower bound is 4DA/(|A|2-1) where DA is the sum of all pairwise Hamming distances in A. Our main theorem establishes identities connecting the number of k-dimensional faces containing exactly e elements of a subset to binomial sums over all subsets of specified cardinality. This yields a parametric family of identities where classical results emerge as special cases. As applications, we derive a geometric interpretation of Vandermonde's identity by examining faces of q-valued cubes, revealing that this classical result naturally arises from counting element distributions. We also obtain a completely new identity for even-weight vectors: (2^(k-1) - 1) times 2^(n-1) times binomial(n,k) equals the sum over i from 1 to floor(n/2) of binomial(n,2i) times binomial(n-2i,k-2i). This identity, valid for all 1 <= k <= n, demonstrates how geometric perspectives can uncover hidden combinatorial relationships. Our framework provides a unified approach for generating new identities and understanding existing ones through subset rank analysis.