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Introducing q-deformed binomial coefficients of words

Antoine Renard, Michel Rigo, Markus A. Whiteland

Abstract

Gaussian binomial coefficients are q-analogues of the binomial coefficients of integers. On the other hand, binomial coefficients have been extended to finite words, i.e., elements of the finitely generated free monoids. In this paper we bring together these two notions by introducing q-analogues of binomial coefficients of words. We study their basic properties, e.g., by extending classical formulas such as the q-Vandermonde and Manvel's et al. identities to our setting. As a consequence, we get information about the structure of the considered words: these q-deformations of binomial coefficients of words contain much richer information than the original coefficients. From an algebraic perspective, we introduce a q-shuffle and a family q-infiltration products for non-commutative formal power series. Finally, we apply our results to generalize a theorem of Eilenberg characterizing so-called p-group languages. We show that a language is of this type if and only if it is a Boolean combination of specific languages defined through q-binomial coefficients seen as polynomials over $\mathbb{F}_p$.

Introducing q-deformed binomial coefficients of words

Abstract

Gaussian binomial coefficients are q-analogues of the binomial coefficients of integers. On the other hand, binomial coefficients have been extended to finite words, i.e., elements of the finitely generated free monoids. In this paper we bring together these two notions by introducing q-analogues of binomial coefficients of words. We study their basic properties, e.g., by extending classical formulas such as the q-Vandermonde and Manvel's et al. identities to our setting. As a consequence, we get information about the structure of the considered words: these q-deformations of binomial coefficients of words contain much richer information than the original coefficients. From an algebraic perspective, we introduce a q-shuffle and a family q-infiltration products for non-commutative formal power series. Finally, we apply our results to generalize a theorem of Eilenberg characterizing so-called p-group languages. We show that a language is of this type if and only if it is a Boolean combination of specific languages defined through q-binomial coefficients seen as polynomials over .
Paper Structure (14 sections, 25 theorems, 117 equations, 4 figures)

This paper contains 14 sections, 25 theorems, 117 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Gaussian binomial coefficients
  3. Binomial coefficients of words
  4. Our contributions
  5. Introducing q-deformations
  6. Combinatorial interpretation
  7. Some classical formulas revisited
  8. Deformations in formal series
  9. Deformed shuffle product
  10. Deformed infiltration
  11. Towards p-groups
  12. Some polynomial algebra
  13. On q-binomials and p-group languages
  14. Conclusions

Key Result

lemma 1

For $u,v \in A^*$, we have that $\binom{u}{v}_{\!\! q}$ is the zero polynomial if and only if $v$ does not appear as a subword in $u$.

Figures (4)

  • Figure 1: Applying recursively the definition \ref{['eq:recdef']} to compute $\binom{0100110}{011}_{\!\! q}$.
  • Figure 2: The multiplication table for the group $\{0,1\}^*/{\sim_{01,q^2+1}}$ with $64$ elements.
  • Figure 3: The structure of the minimal automaton of $L_{01,\mathfrak{R},q^2+1}$.
  • Figure 4: The structure of the minimal automaton of $L_{01,r,2}$.

Theorems & Definitions (61)

  • definition 1
  • lemma 1
  • remark 1
  • theorem 3.1
  • proof
  • corollary 1
  • proof
  • corollary 2
  • proof
  • remark 2
  • ...and 51 more