arXiv subjects

Benjamin Randrianirina

Publications and source records attributed to Benjamin Randrianirina.

3 recordsLinked to original sources

arXiv2026

The Molecular Species $\mathbf{C}_{\alpha}$: Geometric Realization and a Closed Formula for Kronecker Coefficients

In this paper we first introduce the \emph{infinite multi-row periodic pattern of shape $\alpha$} as a geometric realization of $\mathbf{C}_\alpha$. We then give an explicit formula for the coefficients $b^{\lambda}_{\alpha,\beta}$ appearing in the species decomposition \[ \mathbf{C}_\alpha \times \mathbf{C}_\beta = \sum_{\lambda \vdash n} b^\lambda_{\alpha,\beta}\,\mathbf{C}_\lambda. \]

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arXiv2025

Triangular Arrays using context-free grammar

In this work, the Hao grammar $G=\{\, u\rightarrow u^{b_1+b_2+1} v^{a_1+a_2},\quad v\rightarrow u^{b_2}v^{a_2+1} \,\},$ together with the correspondence between grammars and combinatorial differential equations, is employed to obtain an interpretation of any triangular array of the form \[ T(n,k)=(a_2 n + a_1 k + a_0)\,T(n-1,k) + (b_2 n + b_1 k + b_0)\,T(n-1,k-1). \] This lead to have an interpretation of $T(n,k)$ as an increasing tree. Explicit formulas and structural properties are then derived through analytic differential equations. In particular, the $r$-Whitney-Eulerian numbers and the cases where $b_2n+b_1k+b_0=1$ are obtained explicitly. \noindent Applications include new interpretation formulas for the $r$-Eulerian numbers with generating functions. We also obtain full generating functions for the case $a_2=-a_1$ using this approach.

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arXiv2025

Explicit Formulas and Combinatorial Interpretation of Triangular Arrays

Using the lattice paths in $\mathbb{N}\times\mathbb{N}$, we derive a general formula for sequences $\big(T(n,k)\big)$ satisfying the recurrence relation of the form: \begin{equation*} T((n,k)=a_{n,k}T(n-1,k)+b_{n,k}T(n-1,k-1). \end{equation*} We apply this result to the case where $a_{n,k}=a_0+a_1k+a_2n$ and $b_{n,k}=b_0+b_1k+b_2n$. This leads to explicit expressions for $T(n,k)$, with simpler formulas arising in the case $b_2=0$, as well as in the fully general case, using Fa\`a di Bruno's type expression. In particular, we analyze the case $b_{n,k}=1$, which frequently occurs in enumerative combinatorics. Applications include explicit formulas for the $r$-Eulerian numbers.We also express the case $b_{n,k}=1$, using a transition matrix. We apply our results to several sequences. \textbf{Keywords:} triangular recurrence, weighted paths, $r$-Eulerian numbers, combinatorial interpretation.

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