Google Scholar ResearchGate arXiv
Click to expand for abstracts/slides/posters.
[11] The Chromatic Symmetric Function for Unicyclic Graphs (with A. Bingham, L. Johnston, C. Lawson, R. Orellana and C. Sato)
Motivated by the question of which structural properties of a graph can be recovered from the chromatic symmetric function (CSF), we study the CSF of connected unicyclic graphs. While it is known that there can be non-isomorphic unicyclic graphs with the same CSF, we find experimentally that such examples are rare for graphs with up to 17 vertices. In fact, in many cases we can recover data such as the number of leaves, number of internal edges, cycle size, and number of attached non-trivial trees, by extending known results for trees to unicyclic graphs. These results are obtained by analyzing the CSF of a connected unicyclic graph in the star-basis using the deletion-near-contraction (DNC) relation developed by Aliste-Prieto, Orellana and Zamora, and computing the "leading" partition, its coefficient, as well as coefficients indexed by hook partitions. We also give explicit formulas for star-expansions of several classes of graphs, developing methods for extracting coefficients using structural properties of the graph.
[10] Type C K--stanley symmetric functions and Kraskiewicz--Hecke insertion (with J. Arroyo, Z. Hamaker and G. Hawkes)
We study Type C K-Stanley symmetric functions, which are K--theoretic extensions of the Type C Stanley symmetric functions. Our main contribution is Kraśkiewicz--Hecke insertion (KH), a K-theoretic analogue of Kraśkiewicz insertion. Much like Kraśkiewicz insertion enumerates reduced words for signed permutations, KH enumerates their 0--Hecke expressions. The former enumeration witnesses the Type C Stanley expansion into Schur-Q functions. We conjecture that KH extends this to give the Type C K-Stanley expansion into GQ functions, which are K-theory representatives for the Lagrangian Grassmannian introduced by Ikeda and Naruse. We also show Type C K-Stanleys of top fully commutative signed permutations are skew GQ's. This allows us to prove a conjecture of Lewis and Marberg and to give the first conjectural formulas for the expansion of a skew GQ into GQ's. The latter specializes to a rule for multiplying two GQ functions where one has trapezoid shape. This would extend Buch and Ravikumar's Pieri rule, the only known product rule for GQ's.
[9] Hook-valued tableaux uncrowding and tableau switching (with J. Jang, J. S. Kim, J. Pappe and A. Schilling), to appear in SIAM Journal on Discrete Mathematics
Refined canonical stable Grothendieck polynomials were introduced by Hwang, Jang, Kim, Song, and Song. There exist two combinatorial models for these polynomials: one using hook-valued tableaux and the other using pairs of a semistandard Young tableau and (what we call) an exquisite tableau. An uncrowding algorithm on hook-valued tableaux was introduced by Pan, Pappe, Poh, and Schilling. In this paper, we discover a novel connection between the two models via the uncrowding and Goulden--Greene's jeu de taquin algorithms, using a classical result of Benkart, Sottile, and Stroomer on tableau switching. This connection reveals a hidden symmetry of the uncrowding algorithm defined on hook-valued tableaux. As a corollary, we obtain another combinatorial model for the refined canonical stable Grothendieck polynomials in terms of biflagged tableaux, which naturally appear in the characterization of the image of the uncrowding map.
In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan's question, in this paper we define a pattern-avoiding Birkhoff polytope called a c-Birkhoff polytope for each Coxeter element c of the symmetric group. We then show that the c-Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the c-sorting word of the longest permutation. When c=s1s2…sn, this result recovers an affirmative answer to Davis and Sagan's question. Another consequence of this result is that the normalized volume of the c-Birkhoff polytope is the number of the longest chains in the (type A) c-Cambrian lattice.
[7] Top-degree components of Grothendieck and Lascoux polynomials (with T. Yu), Algebraic Combinatorics, Volume 7 (2024) no. 1, pp. 109-135.
The Castelnuovo–Mumford polynomial $\widehat{\mathfrak{G}}_w$ with $w \in S_n$ is the highest homogeneous component of the Grothendieck polynomial $\mathfrak{G}_w$. Pechenik, Speyer and Weigandt define a statistic $\rajcode(\cdot)$ on $S_n$ that gives the leading monomial of $\widehat{\mathfrak{G}}_w$. We introduce a statistic rajcode on any diagram $D$ through a combinatorial construction ``snow diagram'' that augments and decorates $D$. When $D$ is the Rothe diagram of a permutation w, rajcode(D) agrees with the aforementioned $\rajcode(w)$. When D is the key diagram of a weak composition $\alpha$, $\rajcode(D)$ yields the leading monomialof $\widehat{\mathfrak{L}}_\alpha$, the highest homogeneous component of the Lascoux polynomials $\mathfrak{L}_\alpha$. We use $\widehat{\mathfrak{L}}_\alpha$ to construct a basis of $\widehat{V}_n$, the span of $\widehat{\mathfrak{G}}_w$ with $w \in S_n$. Then we show $\widehat{V}_n$ gives a natural algebraic interpretation of a classical q-analogue of Bell numbers.
[6] RSK tableaux and the weak order on fully commutative permutations (with E. Gunawan, H. Russell and B. Tenner), The Electronic Journal of Combinatorics 30(4) (2023)
For each fully commutative permutation, we construct a "boolean core," which is the maximal boolean permutation in its principal order ideal under the right weak order. We partition the set of fully commutative permutations into the recently defined crowded and uncrowded elements, distinguished by whether or not their RSK insertion tableaux satisfy a sparsity condition. We show that a fully commutative element is uncrowded exactly when it shares the RSK insertion tableau with its boolean core. We present the dynamics of the right weak order on fully commutative permutations, with particular interest in when they change from uncrowded to crowded. In particular, we use consecutive permutation patterns and descents to characterize the minimal crowded elements under the right weak order.
[5] Runs and RSK tableaux of boolean permutations (with E. Gunawan, H. Russell and B. Tenner), Annuals of Combinatorics 29, 65–90 (2025)
We define and construct the "canonical reduced word" of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the "run" statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean.
[4] A bijection between K-Kohnert diagrams and reverse set-valued tableaux (with T. Yu), The Electronic Journal of Combinatorics 30(4) (2023)
Lascoux polynomials are K-theoretic analogues of the key polynomials. They both have combinatorial formulas involving tableaux: reverse set-valued tableaux (𝖱𝖲𝖵𝖳) rule for Lascoux polynomials and reverse semistandard Young tableaux (𝖱𝖲𝖲𝖸𝖳) rule for key polynomials. Furthermore, key polynomials have a simple algorithmic model in terms of Kohnert diagrams, which are in bijection with 𝖱𝖲𝖲𝖸𝖳. Ross and Yong introduced K-Kohnert diagrams, which are analogues of Kohnert diagrams. They conjectured a K-Kohnert diagram rule for Lascoux polynomials. We establish this conjecture by constructing a weight-preserving bijection between 𝖱𝖲𝖵𝖳 and K-Kohnert diagrams.
[3] Uncrowding algorithm on hook-valued tableaux (with J. Pappe, W. Poh and A. Schilling), Annals of Combinatorics 26, 261-301 (2022)
Whereas set-valued tableaux are the combinatorial objects associated to stable Grothendieck polynomials, hook-valued tableaux are associated to stable canonical Grothendieck polynomials. In this paper, we define a novel uncrowding algorithm for hook-valued tableaux. The algorithm "uncrowds" the entries in the arm of the hooks and yields a set-valued tableau and a column-flagged increasing tableau. We prove that our uncrowding algorithm intertwines with crystal operators. An alternative uncrowding algorithm that "uncrowds" the entries in the leg instead of the arm of the hooks is also given. As an application of uncrowding, we obtain various expansions of the canonical Grothendieck polynomials.
[2] A crystal on decreasing factorizations in the 0-Hecke monoid (with J. Morse, W. Poh and A. Schilling), The Electronic Journal of Combinatorics 27(2) (2020)
We introduce a type A crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call ⋆-crystal. This crystal is a K-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the ⋆-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.
[1] Virtualization map for the Littelmann path model (with T. Scrimshaw), Transformation Groups (2018) 23: 1045
We show the natural embedding of weight lattices from a diagram folding is a virtualization map for the Littelmann path model, which recovers a result of Kashiwara. As an application, we give a type independent proof that certain Kirillov--Reshetikhin crystals respect diagram foldings, which is a known result on a special case of a conjecture given by Okado, Schilling, and Shimozono.