RANDRIANIRINA

**Abstract:** Using the lattice paths in $\mathbb{N} \times \mathbb{N}$, we derive a general formula for sequences $T(n, k)$ satisfying the recurrence relation of the form: $$T(n, k) = a_{n,k}T(n - 1, k) + b_{n,k}T(n - 1, k - 1)$$ 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 Faà 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. **Keywords:** Combinatorial interpretation; Triangular recurrence; $r$-Eulerian numbers; Weighted paths **2020 Mathematics Subject Classification:** 05A05; 05A15; 05A19

We study two new families of symmetric functions arising from a species-theoretic construction motivated by cycle structure. For each partition of $n$, we define two combinatorial species that decompose into molecules indexed by the same partition, giving rise to two corresponding basis of the homogeneous symmetric functions of degree $n$. We prove that each of these families forms a basis by exhibiting explicit cycle-index formulas and triangular transition matrices to the power-sum basis. Using these constructions, we generalize a classical result describing the Kronecker (Hadamard) product in the homogeneous basis to the two new settings. In particular, we show that the categories generated by these species are closed under the Kronecker product, and that the product of two basis elements expands with nonnegative integer coefficients. Our results provide a new combinatorial framework for studying the Kronecker product and suggest avenues toward interpreting its structure constants

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.