# New combinatorial interpretations of the binomial coefficients

## TL;DR

This paper introduces new combinatorial interpretations of binomial coefficients by linking them to the enumeration of certain pattern-avoiding permutations based on crossings, using generating functions and bijections.

## Contribution

It provides novel combinatorial interpretations of binomial coefficients through pattern-avoiding permutations and introduces a q-tableau counting these permutations by crossings.

## Key findings

- Binomial coefficients count (123,132) and (123,213)-avoiding permutations by crossings.
- A q-tableau of powers of two counts (213,312) and (132,312)-avoiding permutations by crossings.
- Generating functions and bijections are used to establish these combinatorial interpretations.

## Abstract

Using generating functions and some trivial bijections, we show in this paper that the binomial coefficients count the set of (123,132) and (123,213)-avoiding permutations according to the number of crossings. We also define a q-tableau of power of two and prove that it counts the set of (213,312) and (132,312)-avoiding permutations according to the number of crossings.