The (P, omega)-partition generating function K(P,omega)(x) is a quasisymmetric function obtained from a labeled poset. Recently, Liu and Weselcouch gave a formula for the coefficients of K(P,omega)(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, omega)-partitions and provide a short combinatorial proof, avoiding the Hopf algebra machinery used by Liu-Weselcouch. (c) 2026 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

We prove that monomial coefficients in affine families of key and Schubert polynomials are eventually polynomial. The proof combines Demazure operators with vector partition functions and, in the Schubert case, P. Magyar's orthodontic formula. These coefficient results extend to finite products. Using M. Watanabe's Schubert duality, we deduce that stretched Schubert structure constants are eventually polynomial, proving a conjecture of I. Pak and Z. Slonim. For key polynomials, this resolves the polynomiality part of a conjecture of P. Alexandersson and E. Alhajjar.

A linear differential operator T=Q(z)d/dz+P(z) with polynomial coefficients defines a continuous family of Hutchinson operators when acting on the space of positive powers of linear forms. In this context, T has a unique minimal Hutchinson-invariant set M_CH^T in the complex plane. Using a geometric interpretation of its boundary in terms of envelopes of certain families of rays, we subdivide this boundary into local and global arcs (the former being portions of integral curves of the rational vector field Q(z)/P(z)∂_z), and singular points of different types which we classify below. The latter decomposition of the boundary of M_CH^T is largely determined by its intersection with the plane algebraic curve formed by the inflection points of trajectories of the field Q(z)/P(z)∂_z. We provide an upper bound for the number of local arcs in terms of degrees of P and Q. As an application of our classification, we obtain a number of global geometric properties of minimal Hutchinson-invariant sets.

We study six Eulerian-type polynomial families. We prove that the descent polynomials of derangements are real-rooted, settling the derangement part of a conjecture of S. Fu, Z. Lin, and J. Zeng. The proof uses a compatible-pair recursion and finite-symbol stability. We also resolve the real-rootedness conjecture in OEIS A335340, strengthen the known rowwise real-rootedness of an even-top descent family to consecutive strict interlacing, and prove real-rootedness, consecutive interlacing, and real-rooted gamma-polynomials for U. Shankar's super-Eulerian polynomials. A differential recurrence gives consecutive weak interlacing for ternary words counted by increasing runs. Finally, we prove stability of the peak-value refinement and consecutive interleaving of its positive weighted diagonals, settling a conjecture of P. Alexandersson and O. Nabawanda.

We prove real-rootedness of the Chow polynomials of the noncrossing partition lattices by transferring tieless parking functions to finite-alphabet Smirnov words and applying a last-letter interlacing recurrence. We also derive a triangular recurrence for peaks and ties and identify the peakless-tieless descent polynomial as the Narayana polynomial. For the toric g-contributions of Ehrenborg–Hetyei–Readdy, we exhibit a fixed-row common interlacer and establish real-rootedness of all nonnegative row sums. Individual real-rootedness follows in particular; Q. Xiao recently proved it independently by a different differential recurrence. We also give a separate finite Schur–Szegő convolution proof of the individual statement. These results prove Conjecture 4.2 of Xiao and Conjecture 11.2 of Ehrenborg–Hetyei–Readdy, with consequences for weakly 123-avoiding parking functions. We also prove real-rootedness for the image-size polynomial on all parking functions and for the ascent and descent polynomials of four two-pattern-avoiding classes.

We describe an efficient method for computing the Ehrhart polynomial of Gelfand–Tsetlin polytopes arising from Kostka coefficients. The key idea is to exploit Ehrhart–Macdonald reciprocity: evaluating the Ehrhart polynomial at negative integers reduces to counting strict Gelfand–Tsetlin patterns, which are often zero or very small for low dilations. Combined with an adaptive strategy that chooses the cheapest evaluation point (positive or negative) at each step, this yields substantial practical speedups compared to general-purpose polytope software. We benchmark against 𝙾𝚂𝙲𝙰𝚁/𝚙𝚘𝚕𝚢𝚖𝚊𝚔𝚎, and illustrate the broader applicability of the method through order polytopes and permutation posets. The implementation is available in the Rust package, with related optimizations also incorporated in the new replacement for .

We introduce rook-Eulerian polynomials, a generalization of the classical Eulerian polynomials arising from complete rook placements on Ferrers boards, and prove that they are real-rooted. We show that a natural context in which to interpret these rook placements is as lower intervals of 312-avoiding permutations in the Bruhat order. We end with some variations and generalizations along this theme.

Given a linear ordinary differential operator T with polynomial coefficients, we study the class of closed subsets of the complex plane such that T sends any polynomial (respectively, any polynomial of degree exceeding a given positive integer) with all roots in a given subset to a polynomial with all roots in the same subset or to 0 . Below we discuss some general properties of such invariant subsets, as well as the problem of existence of the minimal under inclusion invariant subset.

We consider Jack polynomials J_λ and their shifted analogue J^#_λ. In 1989, Stanley conjectured that ⟨ J_μJ_ν, J_λ⟩ is a polynomial with nonnegative coefficients in the parameter α. In this note, we extend this conjecture to the case of shifted Jack polynomials.

We define and study rook matroids, the bases of which correspond to non-nesting rook placements on a skew Ferrers board. We show that rook matroids are a subclass of both transversal matroids and positroids; they also bear a subtle relationship to lattice path matroids that centers around not having the quaternary matroid Q_6 as a minor. The enumerative and distributional properties of non-nesting rook placements stand in contrast to those of usual rook placements: the non-nesting rook polynomial is not real-rooted in general, and is instead ultra-log-concave. We leverage this property together with a correspondence between rook placements and linear extensions of a poset to show that if P is a naturally labeled width two poset, then the P-Eulerian polynomial W_P is ultra-log-concave. This takes an important step towards resolving a log-concavity conjecture of Brenti (1989) and completes the story of the Neggers–Stanley conjecture for naturally labeled width two posets.

This paper, being the sequel of [An inverse problem in Polya-Schur theory. I. Non-genegerate and degenerate operators], studies a class of linear ordinary differential operators with polynomial coefficients called exactly solvable; such an operator sends every polynomial of sufficiently large degree to a polynomial of the same degree. We focus on invariant subsets of the complex plane for such operators when their action is restricted to polynomials of a fixed degree and discover a connection between this topic and classical complex dynamics and its multi-valued counterpart. As a very special case of invariant sets we recover the Julia sets of rational functions.

In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in details in the simplest case of operators of order $1$. Namely, assuming that such an operator $T$ has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of $T$, we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any $T$ with non-constant coefficients, there exists a unique minimal under inclusion invariant set $\mathrm{M}^T_{CH}$ and find explixitly when it equals $\mathbb{C}$.

We give a bijective proof of a result by R. Mantaci and F. Rakotondrajao from 2003, regarding even and odd derangement with a fixed number of excedances. We refine this result by also considering the set of right-to-left minima.

We generalize several classical results about Schur functions to the family of cylindric Schur functions. First, we give a combinatorial proof of a Murnaghan--Nakayama formula for expanding cylindric Schur functions in the power-sum basis. We also explore some cases where this formula is cancellation-free. The second result is polynomiality of Kostka coefficients associated with stretched row-flagged skew Schur functions. This implies polynomiality of stretched cylindric Kostka coefficients. This generalizes a result by E. Rassart from 2004. Finally, we also show the saturation property for the row-flagged skew Kostka coefficients which also implies the saturation property for cylindric Schur functions.

In this note, we provide a short proof of Theorem 4.3 in the paper titled Crystals, semistandard tableaux and cyclic sieving phenomenon , by Y.-T. Oh and E. Park, which concerns a cyclic sieving phenomenon on semi-standard Young tableaux. We also extend their result to skew shapes.

We study a subset of permutations, where entries are restricted to having the same remainder as the index, modulo some integer $k \geq 2$. We show that when also imposing the classical 132- or 213-avoidance restriction on the permutations, we recover the Fuss--Catalan numbers and some special cases of the Raney numbers. Surprisingly, an analogous statement also holds when we impose the mod $k$ restriction on a Catalan family of subexcedant functions. Finally, we completely enumerate all combinations of mod-$k$-alternating permutations, avoiding two patterns of length 3. This is analogous to the systematic study by Simion and Schmidt, of permutations avoiding two patterns of length 3.

We make progress towards understanding the structure of Littlewood-Richardson coefficients $g_{\lambda,\mu}^{\nu}$ for products of Jack symmetric functions. Building on recent results of the second author, we are able to prove new cases of a conjecture of Stanley in which certain families of these coefficients can be expressed as a product of upper or lower hook lengths for every box in each of the partitions. In particular, we prove that conjecture in the case of a rectangular union, i.e. for $g_{\mu,\bar \sigma}^{\mu \cup m^n}$ where $\bar \sigma$ is the complementary partition of $\sigma = \mu \cap m^n$ in the rectangular partition $m^n$. We give a formula for these coefficients through an explicit prescription of such choices of hooks. Lastly, we conjecture an analogue of this conjecture of Stanley holds in the case of Shifted Jack functions.

We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the "longest increasing subsequence" have Ehrhart quasi-polynomials which are honest polynomials, even though they are just rational polytopes in general. We do this by defining a continuous, piecewise-linear bijection to a certain Gelfand-Tsetlin polytope. This bijection is not an integral equivalence but it respects lattice points in the appropriate way to imply that the two polytopes have the same Ehrhart (quasi-)polynomials. In fact, the bijection is essentially the Robinson-Schensted-Knuth correspondence.

We give a new characterization of the vertical-strip LLT polynomials GP (x; q) as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if q is replaced by q + 1, thus recovering a recent result of M. D’Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D’Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.

We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type A and type B. Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.