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Binomial determinants: some closed formulae

Laura González, Francesc Planas-Vilanova

Abstract

This paper is intended to give closed formulae for binomial determinants with consecutive or almost consecutive rows or columns, as well as calculating the generator of left nullspaces defined by some binomial matrices. In the meantime, we reprove, by different means, the positivity of binomial determinants shown by Gessel and Viennot.

Binomial determinants: some closed formulae

Abstract

This paper is intended to give closed formulae for binomial determinants with consecutive or almost consecutive rows or columns, as well as calculating the generator of left nullspaces defined by some binomial matrices. In the meantime, we reprove, by different means, the positivity of binomial determinants shown by Gessel and Viennot.

Paper Structure

This paper contains 8 sections, 20 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Size reduction and positivity
  3. A closed formula for consecutive columns
  4. A closed formula for consecutive rows
  5. Consecutive rows and almost consecutive columns
  6. Almost consecutive rows and consecutive columns
  7. Rank and left nullspaces of some binomials matrices
  8. Interchanging rows and columns

Key Result

Lemma 2.2

Let $I=\{i_1,\ldots,i_d\}$ and $J=\{j_1,\ldots,j_d\}$.

Theorems & Definitions (50)

  • Lemma 2.2: Reduction to $J<I$, $j_1=0$
  • proof
  • Theorem 2.3: Size reduction
  • proof
  • Example 2.4
  • Corollary 2.5: Positivity
  • proof
  • Corollary 2.6
  • proof
  • Theorem 3.1: Consecutive columns
  • ...and 40 more