Abstract
We prove a $q$-analogue of the modular hook length formula using position sequences. These position sequences, which correspond to moving the beads in a mathematical abacus, provide a new combinatorial interpretation for the characters of the irreducible representations of the symmetric group.
Highlights
Let n and k be positive integers and let λ and μ = (μ1, μ2, . . . ) be integer partitions of n
A rim hook of length k is a sequence of k connected cells in the (English) Young diagram for λ that begins in a cell on the southeast boundary and travels up along the southeast edge such that its removal leaves the Young diagram of a smaller integer partition
If p = p1 · · · p is any sequence of integers, the major index of p, denoted maj p, is equal to i where the sum runs over all indices the electronic journal of combinatorics 26(4) (2019), #P4.18 i such that pi > pi+1
Summary
If p = p1 · · · p is any sequence of integers, the major index of p, denoted maj p, is equal to i where the sum runs over all indices the electronic journal of combinatorics 26(4) (2019), #P4.18 i such that pi > pi+1 Adapting this idea for standard tableaux, if T ∈ RHT(λ1,...,1) is a standard tableau, the integer i is a descent in T if rim hook i appears in a row above that of rim hook i + 1. The second generalization of the hook length formula involves rim hook tableaux of shape λ and content
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
