Sivaramakrishnan Sivasubramanian

Reifegerste and independently, Petersen and Tenner studied a statistic $\mathrm{drops}()$ on permutations in $\mathfrak{S}_n$. Two other studied statistics on $\mathfrak{S}_n$ are $\mathrm{depth}$ and $\mathrm{exc}$. Using descents in ${\it canonical\ reduced\ words}$ of elements in $\mathfrak{S}_n$, we give an involution $f_A: \mathfrak{S}_n \mapsto \mathfrak{S}_n$ that leads to a neat formula for the signed trivariate enumerator of $\mathrm{drops},\mathrm{depth}, \mathrm{exc}$ in $\mathfrak{S}_n$. This gives a simple formula for the signed univariate drops enumerator in $\mathfrak{S}_n$. For the type-B Coxeter group $\mathfrak{B}_n$ as well, using similar techniques, we show analogous results. For the type D Coxeter group, we again get analogous results, but our proof is inductive. Under the famous Foata-Zeilberger bijection $φ_{FZ}$ which takes permutations to restricted Laguerre histories, we show that permutations $π$ and $f_A(π)$ map to the same Motzkin path, but have different history components. Using the Foata-Zeilberger bijection, we also get a continued fraction for the generating function enumerating the pair of statistics $\mathrm{drops}$ and $\mathrm{MAD}$. Graham and Diaconis determined the mean and the variance of the Spearman metric of disarray $D(π)$ when one samples $π$ from $\mathfrak{S}_n$ at random. As an application of our results, we get the mean and variance of the statistic $\mathrm{drops}(π)$ when we sample $π$ from $\mathcal{A}_n$ at random.

The Four point condition (abbreviated as 4PC) is a condition used to test if a given distance matrix arises from shortest path distances on trees. From a tree \(T\), Bapat and Sivasubramanian defined a matrix \(\operatorname{Min4PC}_T\) based on this condition. They also gave a basis \(B\) for the row space of \(\operatorname{Min4PC}_T\) and determined its Smith Normal Form. In this paper, we consider the matrix \(\operatorname{Min4PC}_T[B,B]\) restricted to a basis \(B\) and give an explicit inverse for it. It is known that the distance matrix \(D_T\) of a tree \(T\), is invertible and that its inverse is a rank-one update of its scaled Laplacian. Our inverse has a similar form and is a rank-one update of a Laplacian like matrix.

Let $T$ be a tree on $n$ vertices and let $L_q^T$ be the $q$-analogue of its Laplacian. For a partition $λ\vdash n$, let the normalized immanant of $L_q^T$ indexed by $λ$ be denoted as $d_λ(L_q^T)$. A string of inequalities among $d_λ(L_q^T)$ is known when $λ$ varies over hook partitions of $n$ as the size of the first part of $λ$ decreases. In this work, we show a similar sequence of inequalities when $λ$ varies over two row partitions of $n$ as the size of the first part of $λ$ decreases. Our main lemma is an identity involving binomial coefficients and irreducible character values of $S_n$ indexed by two row partitions. Our proof can be interpreted using the combinatorics of Riordan paths and our main lemma admits a nice probabilisitic interpretation involving peaks at odd heights in generalized Dyck paths or equivalently involving special descents in Standard Young Tableaux with two rows. As a corollary, we also get inequalities between $d_{λ_1}(L_q^{T_1})$ and $d_{λ_2}(L_q^{T_2})$ when $T_1$ and $T_2$ are comparable trees in the $GTS_n$ poset and when $λ_1$ and $λ_2$ are both two rowed partitions of $n$, with $λ_1$ having a larger first part than $λ_2$.

Carlitz and Scoville in 1973 considered a four variable polynomial that enumerates permutations in $\mathfrak{S}_n$ with respect to the parity of its descents and ascents. In recent work, Pan and Zeng proved a $q$-analogue of Carlitz-Scoville's generating function by enumerating permutations with the above four statistice along with the inversion number. Further, they also proved a type B analogue by enumerating signed permutations with respect to the parity of descents and ascents. In this work we prove a $q$-analogue of the type B result of Pan and Zeng by enumerating permutations in $\mathfrak{B}_n$ with the above four statistics and the type B inversion number. We also obtain a $q$-analogue of the generating function for the type B bivariate alternating descent polynomials. We consider a similar five-variable polynomial in the type D Coxeter groups as well and give their egf. Alternating descents for the type D groups were previously also defined by Remmel, but our definition is slightly different. As a by-product of our proofs, we get bivariate $q$-analogues of Hyatt's recurrences for the type B and type D Eulerian polynomials. Further corollaries of our results are some symmetry relations for these polynomials and $q$-analogues of generating functions for snakes of types B and D.