Jae-Seong Oh

Abstract We introduce a new class of permutations, called web permutations. Using these permutations, we provide a combinatorial interpretation for entries of the transition matrix between the Specht and $\operatorname {SL}_2$ -web bases of the irreducible $ \mathfrak {S}_{2n} $ -representation indexed by $ (n,n) $ , which answers Rhoades’s question. Furthermore, we study enumerative properties of these permutations.

We introduce a new class of permutations, called web permutations. Using these permutations, we provide a combinatorial interpretation for entries of the transition matrix between the Specht and web bases, which answers Rhoades's question. Furthermore, we study enumerative properties of these permutations.

Abstract In this paper, we introduce the $\alpha $-chromatic symmetric functions $\chi ^{(\alpha )}_\pi [X;q]$, extending Shareshian and Wachs’ chromatic symmetric functions with an additional real parameter $\alpha $. We present positive combinatorial formulas with explicit interpretations. Notably, we show an explicit monomial expansion in terms of the $\alpha $-binomial basis and an expansion into certain chromatic symmetric functions in terms of the $\alpha $-falling factorial basis. Among v

We define the acyclic orientation polynomial of a graph to be the generating function for the sinks of its acyclic orientations. Stanley proved that the number of acyclic orientations is equal to the chromatic polynomial evaluated at $-1$ up to sign. Motivated by this link between acyclic orientations and the chromatic polynomial, we develop "acyclic orientation" analogues of theorems concerning the chromatic polynomial of Birkhoff, Whitney, and Greene-Zaslavsky. As an application, we provide a

The Garsia--Haiman module is a bigraded $\mathfrak{S}_n$-module whose Frobenius image is a Macdonald polynomial. The method of orbit harmonics promotes an $\mathfrak{S}_n$-set $X$ to a graded polynomial ring. The orbit harmonics can be applied to prove cyclic sieving phenomena which is a notion that encapsulates the fixed-point structure of finite cyclic group action on a finite set. By applying this idea to the Garsia--Haiman module, we provide cyclic sieving results regarding the enumeration o

We initiate the study of the Macdonald intersection polynomials $\operatorname{I}_{μ^{(1)},\dots,μ^{(k)}}[X;q,t]$, which are indexed by $k$-tuples of partitions $μ^{(1)},\dots,μ^{(k)}$. These polynomials are conjectured to be equal to the bigraded Frobenius characteristic of the intersection of Garsia-Haiman modules, as proposed by the science fiction conjecture of Bergeron and Garsia. In this work, we establish the vanishing identity and the shape independence of the Macdonald intersection poly

We prove that the symmetric function $e_{(1^k)}[-MX^{m,n}] \cdot 1$, arising from the elliptic Hall algebra, equals the generating function for $k$-tuples of cyclic $(m,n)$-parking functions. This result resolves a conjecture of Gorsky--Mazin--Vazirani and Wilson, establishing that the elliptic Hall algebra governs the Khovanov--Rozansky homology of torus links $T(km,kn)$. Consequently, this provides an affirmative answer to a question of Galashin and Lam in the torus link case. As a key step in

Abstract The celebrated Haglund–Haiman–Loehr (HHL) formula provides an explicit monomial expansion of the Macdonald polynomials. In 1994, Butler introduced a refinement of the Macdonald polynomial and conjectured its Schur positivity. According to the Science Fiction conjecture by Bergeron and Garsia, this refinement represents the “intersection” of Macdonald polynomials. In this work, we introduce a novel combinatorial tool, the column exchange rule , which enables us to derive a positive monom

Abstract Let be a matrix of variables and let be the polynomial ring in these variables. Given two weak compositions of lengths and , we study the ideal generated by row sums, column sums, monomials in row of degree , and monomials in column of degree . We prove results connecting algebraic properties of the quotient ring with the set of ‐contingency tables. The standard monomial basis of with respect to a diagonal term order is encoded by the matrix‐ball avatar of the Robinson–Schensted–Knuth c