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Cohen-Macaulay types of certain edge subrings of bipartite graphs and (generalized) Fuss-Catalan numbers

Alin Ştefan

Abstract

We give an example of two non isomorphic coordinate rings of a special kind of convex polyominoes whose Cohen-Macaulay types are generalized Fuss-Catalan numbers. We further provide a determinantal formula for these numbers.

Cohen-Macaulay types of certain edge subrings of bipartite graphs and (generalized) Fuss-Catalan numbers

Abstract

We give an example of two non isomorphic coordinate rings of a special kind of convex polyominoes whose Cohen-Macaulay types are generalized Fuss-Catalan numbers. We further provide a determinantal formula for these numbers.
Paper Structure (6 sections, 7 theorems, 32 equations)

This paper contains 6 sections, 7 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Prelimininaries
  3. Cones
  4. Polyominoes
  5. Simple paths with general bounderies
  6. The Main result

Key Result

Theorem 1

Let $\mathcal{P}$ be a convex polyomino. Then $K[\mathcal{P}]$ is a normal Cohen-Macaulay domain of dimension $|V(\mathcal{P})|$ - $|\mathcal{P}|$.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Proposition 4
  • proof
  • Corollary 5
  • proof
  • Example 6
  • Lemma 7
  • proof
  • ...and 4 more