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Identities that represent powers of positive integers using multinomial coefficients

Shoichi Kamada

Abstract

In this paper, we show combinatorial identities that represent powers of positive integers using multinomial coefficients, which do not come from the multinomial theorem and the multinomial Vandermonde's convolution.

Identities that represent powers of positive integers using multinomial coefficients

Abstract

In this paper, we show combinatorial identities that represent powers of positive integers using multinomial coefficients, which do not come from the multinomial theorem and the multinomial Vandermonde's convolution.
Paper Structure (8 sections, 6 theorems, 32 equations)

This paper contains 8 sections, 6 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Contribution
  4. Identities using sums of multinomial coefficients
  5. An identity using a sum of products of two multinomial coefficients
  6. An identity where $K$ is a group
  7. Examples
  8. Concluding remarks

Key Result

Theorem 2.1

For any positive integers $m$ and $s$ and any nonempty set $K$, there are a positive integer $n$ and an ${\mathcal{S}}_{n}$-invariant subset $V\subseteq K^{n}$ with $|V|=s$ such that

Theorems & Definitions (12)

  • Theorem 2.1
  • proof : Proof of Theorem \ref{['thm:col_identity']}
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['thm:row_f_identity']}
  • Theorem 3.1
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['lem:implication_between_equivalence_relations']}
  • proof : Proof of Theorem \ref{['thm:row_col_identity']}
  • Theorem 4.1
  • Lemma 4.2
  • ...and 2 more