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Fuss-Catalan Triangles

Francesca Aicardi

Abstract

For each $p>0$ we define by recurrence a triangle $T^p(n,k)$ whose rows sum to the Fuss-Catalan numbers $ \frac{1}{p n+1}\binom{pn+1}{n}$, generalizing the known Catalan triangle corresponding to the case $p=2$. (In fact, $T^p(n,k)$ has an explicit formula counting simple lattice paths). Moreover, for some small values of $p$, the signed sums turn out to be known sequences. \end{abstract}

Fuss-Catalan Triangles

Abstract

For each we define by recurrence a triangle whose rows sum to the Fuss-Catalan numbers , generalizing the known Catalan triangle corresponding to the case . (In fact, has an explicit formula counting simple lattice paths). Moreover, for some small values of , the signed sums turn out to be known sequences. \end{abstract}
Paper Structure (5 sections, 3 theorems, 20 equations, 2 figures)

This paper contains 5 sections, 3 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fuss-Catalan triangles
  3. Some applications
  4. Fuss-Catalan triangles and double planar partitions.
  5. Signed sums

Key Result

Proposition 1

The integers $T^p(n,k)$ can be defined for $n\ge 0$, $1-p \le k \le n$ by the initial conditions E0 together with and the recurrence for $n>0$ and $0\le k<n$:

Figures (2)

  • Figure 1: A diagram with arcs and ties representing a double partition of $[10]$ with 3 boxes
  • Figure 2: From a diagram in $\mathcal{P}\mathcal{P}_{n-1}$ with $j=5$ boxes we get $j-k+2=4$ diagrams in $\mathcal{P}\mathcal{P}_n$ with $k=3$ boxes

Theorems & Definitions (9)

  • Definition 1
  • Proposition 1
  • proof
  • Remark 1
  • Proposition 2
  • Remark 2
  • Remark 3
  • Proposition 3
  • proof