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A weighted Murnaghan-Nakayama rule for $(P, w)$-partitions

Per Alexandersson, Olivia Nabawanda

TL;DR

The paper addresses the problem of expressing the weighted $(P,\omega)$-partition generating function $K^d_{(P,\omega)}(\mathbf{x})$ in the quasisymmetric power-sum basis, providing a Murnaghan--Nakayama-type rule for weighted labeled posets. It develops a purely combinatorial proof that leverages two recurrences and an induction reducing to naturally labeled weighted chains, with a probabilistic base-case that connects to linear extensions of certain trees. The main result is a unifying formula (Theorem main) for the $\hat{\psi}$-expansion, which recovers and extends the prior expansions for natural and arbitrary labelings without Hopf-algebra machinery. The work also clarifies border-strip interpretations, offers a new combinatorial path to classic MN-type rules, and raises open questions about cylindrical posets, cancellation behavior for rectangular shapes, and potential extensions to set-valued partitions.

Abstract

The $(P, w)$-partition generating function $K_{(P,w)}(x)$ is a quasisymmetric function obtained from a labeled poset. Recently, Liu and Weselcouch gave a formula for the coefficients of $K_{(P,w)}(x)$ when expanded in the quasisymmetric power sum function basis. This formula generalizes the classical Murnaghan--Nakayama rule for Schur functions. We extend this result to weighted $(P, w)$-partitions and provide a short combinatorial proof, avoiding the Hopf algebra machinery used by Liu--Weselcouch.

A weighted Murnaghan-Nakayama rule for $(P, w)$-partitions

TL;DR

The paper addresses the problem of expressing the weighted -partition generating function in the quasisymmetric power-sum basis, providing a Murnaghan--Nakayama-type rule for weighted labeled posets. It develops a purely combinatorial proof that leverages two recurrences and an induction reducing to naturally labeled weighted chains, with a probabilistic base-case that connects to linear extensions of certain trees. The main result is a unifying formula (Theorem main) for the -expansion, which recovers and extends the prior expansions for natural and arbitrary labelings without Hopf-algebra machinery. The work also clarifies border-strip interpretations, offers a new combinatorial path to classic MN-type rules, and raises open questions about cylindrical posets, cancellation behavior for rectangular shapes, and potential extensions to set-valued partitions.

Abstract

The -partition generating function is a quasisymmetric function obtained from a labeled poset. Recently, Liu and Weselcouch gave a formula for the coefficients of when expanded in the quasisymmetric power sum function basis. This formula generalizes the classical Murnaghan--Nakayama rule for Schur functions. We extend this result to weighted -partitions and provide a short combinatorial proof, avoiding the Hopf algebra machinery used by Liu--Weselcouch.
Paper Structure (14 sections, 10 theorems, 49 equations, 7 figures)

This paper contains 14 sections, 10 theorems, 49 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Labeled posets and order-preserving maps
  4. Quasisymmetric functions
  5. Weighted (P,omega)-partitions
  6. Quasisymmetric power sums
  7. (P,w)-partitions and power sums
  8. The classical Murnaghan--Nakayama rule
  9. A weighted generalization
  10. Splitting the weights
  11. Naturally labeled weighted chains
  12. Open problems
  13. Comparison with the cylindrical Murnaghana--Nakayama rule
  14. Rectangular compositions and cancellation-free sums

Key Result

Lemma 5

Suppose $(P, \omega,d)$ is a weighted poset where $a,b \in P$ are incomparable and $\omega(a)<\omega(b)$. Let $P'$ and $P"$ be obtained from $P$ by adding the relation $a <_{P'} b$ and $a>_{P"} b$, respectively. Then the generating function $K_{(P,\, \omega)}^d(\mathbf{x})$ satisfies the recurrence

Figures (7)

  • Figure 1: Splitting a vertex with weight $d_1+d_2$ into two smaller vertices.
  • Figure 2: The step-by-step bijection from $\mathcal{O}^*(P', \omega)$ (left) to $\mathcal{O}^*(P", \omega)$ (right) for the cases where removal of $(a,b)$ results in two roots (the two middle posets). All edges except $(a,b)$ are optional and do not have any effect, but the type (strict/weak) must be as indicated. Note, $\mathfrak{t}(b)=1$ in the leftmost poset, but $\mathfrak{t}(b)=-1$ in the rightmost.
  • Figure 3: Two examples of staircase diagrams illustrating sets of events in the probability space $\Omega$. The first diagram represents $d_1^2\cdot d_2^2 \cdot d_3^2 \cdot d_6$ different integer vectors in $\Omega$.
  • Figure 4: The staircase diagrams belonging to $\Omega_{(2,3,2)}$ must have the marked boxes selected, but we are free to chose any box in each shaded column.
  • Figure 5: The staircase diagrams for $\Omega_{(3)}$, $\Omega_{(1, 2)}$, $\Omega_{(2, 1)}$, $\Omega_{(1, 1, 1)}$ respectively.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Definition 1
  • Example 2
  • Definition 3
  • Example 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Remark 7
  • Definition 8
  • ...and 25 more