A combinatorial proof of a plethystic Murnaghan--Nakayama rule

This article gives a combinatorial proof of a plethystic generalization of the Murnaghan--Nakayama rule. The main result expresses the product of a Schur function with the plethysm $p_r \circ h_n$ as an integral linear combination of Schur functions. The proof uses a sign-reversing involution on sequences of bead moves on James' abacus, inspired by the arguments in N. Loehr, Abacus proofs of Schur function identities, SIAM J. Discrete Math. 24 (2010), 1356-1370.Comment: 9 pages, 2 figures, expanded and revised versio

On the complexity of computing Kronecker coefficients

We study the complexity of computing Kronecker coefficients $g(\lambda,\mu,\nu)$. We give explicit bounds in terms of the number of parts $\ell$ in the partitions, their largest part size $N$ and the smallest second part $M$ of the three partitions. When $M = O(1)$, i.e. one of the partitions is hook-like, the bounds are linear in $\log N$, but depend exponentially on $\ell$. Moreover, similar bounds hold even when $M=e^{O(\ell)}$. By a separate argument, we show that the positivity of Kronecker coefficients can be decided in $O(\log N)$ time for a bounded number $\ell$ of parts and without restriction on $M$. Related problems of computing Kronecker coefficients when one partition is a hook, and computing characters of $S_n$ are also considered.Comment: v3: incorporated referee's comments; accepted to Computational Complexit

A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type

We consider the Grothendieck polynomials appearing in the K-theory of Grassmannians, which are analogs of Schur polynomials. This paper aims to establish a version of the Murnaghan-Nakayama rule for Grothendieck polynomials of the Grassmannian type. This rule allows us to express the product of a Grothendieck polynomial with a power sum symmetric polynomial into a linear combination of other Grothendieck polynomials.Comment: 10 pages, 7 figure

Murnaghan-Nakayama Rule The Explanation and Usage of the Algorithm

Character values are not the easiest to calculate, so it is important to find good algorithms that can help ease these calculations. In the 20th century, the two mathematicians Murnaghan and Nakayama developed a rule that calculates character values for partitions on some computations. This rule has later been given the name The Murnaghan-Nakayama rule, after these two authors. The Murnaghan-Nakayama rule is a combinatorial method for computing character values of irreducible representations of symmetric groups. This makes this rule an important part of representation theory. One of the versions of this rule is stated in the recursive Murnaghan-Nakayama rule. Where, in this version, we can use border strips and diagrams to calculate the character values of representations on a given composition. This algorithm is quite fast in these calculations. The Murnaghan-Nakayama rule can also be considered a central algorithm in representation theory over symmetric groups. It is a fascinating and powerful algorithm that has a strong connection to both combinatorics and representation theory

The combinatorics of Jeff Remmel

We give a brief overview of the life and combinatorics of Jeff Remmel, a mathematician with successful careers in both logic and combinatorics