arXiv subjects

M. Parvathi

Publications and source records attributed to M. Parvathi.

5 recordsLinked to original sources

arXiv2026

p^(k)-Fibonacci Numbers of the p-Bratteli Diagram for Every Odd Prime p and Integer k>=0

We study paths in the p-Bratteli diagram associated with hook partitions, where p is an odd prime. By comparing blocks along a path, we define inversions and descents. We prove that the sign balance derived from inversions vanishes at every vertex of the diagram. Using descents, we introduce the p^(k)-Fibonacci numbers and derive recurrence relations for them. For k=0, we recover the OEIS sequence A391520, while for k>=1 we obtain new families of Fibonacci-type sequences.

math.CO↗

arXiv2025

Study of $p$-Young tableaux, Robinson-Schensted correspondence and the lacunary Cauchy identity of group algebras $KG_{r}$ and $KSG_{r}$

In this paper, we develop the Robinson-Schensted correspondence between the elements of the groups $G_{r}$ $(\mathbb{Z}_{p^{r}}\rtimes \mathbb{Z}^{*}_{p^{r}})$ and $SG_{r}$ $(\mathbb{Z}_{p^{r-1}}\rtimes \mathbb{Z}^{*}_{p^{r}})$, along with a pair of the standard $p$-Young tableaux. This approach differs from the classical method, and ours is based on matrix units arising from orthogonal primitive idempotents computed for every group algebra. Some classical properties of the Robinson-Schensted correspondence are discussed. As a by-product, we also extend the Cauchy identity to our setup, which we refer to as the lacunary Cauchy identity. This study offers new insights into the representation theory of these groups and their combinatorial structures.

math.RT↗

arXiv2025

Study of $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers for every odd prime $p$ and $k\geq0$

In this paper, we define the notion of descent for the paths in the $p$-Bratteli diagram. This leads to the definition of $p^{k}$-Eulerian polynomials, whose coefficients count the number of paths with a given number of descents. We provide a method for constructing the $p^{k}$-Eulerian polynomials at each vertex. Furthermore, we compute the total number of descents of all paths ending at a given vertex as the corresponding $p^{k}$-Fibonacci numbers. We show that the derivative of the $p^{k}$-Eulerian polynomial evaluated at 1 for a fixed vertex equals the corresponding $p^{k}$-Fibonacci number. Finally, we discuss the generating function for the sequence of $p^{k}$-Fibonacci numbers and the recurrence relations they satisfy.

math.CO↗

arXiv2015

Gram Matrices and Stirling numbers of a class of Diagram Algebras

In this paper, we introduce Gram matrices for the signed partition algebras, the algebra of $\mathbb{Z}_2$-relations and the partition algebras. We prove that the Gram matrix is similar to a matrix which is a direct sum of block submatrices. In this connection, $(s_1, s_2, r_1, r_2, p_1, p_2)$-Stirling numbers of the second kind are introduced and their identities are established. As a consequence, the semisimplicity of a signed partition algebra is established.

math.RA↗

arXiv2015

Eigenvalues of Gram Matrices of a class of Diagram Algebras

In this paper, we introduce symmetric diagram matrices $A_{s+r,s}$ of size ${_{(s+r)}}C_s$ whose entries are $\{x_i\}_{min\{s,r\}}$. We compute the eigenvalues of symmetric diagram matrices using elementary row and column operations inductively. As a byproduct, we obtain the eigenvalues of Gram matrices of a larger class of diagram algebras like the signed partition algebras, algebra of $\mathbb{Z}_2$ relations and partition algebras.

math.RA↗