<   AlCoVE: an Algebraic Combinatorics Virtual Expedition   >

On the Rational Generating Function for Intervals of Partitions
S M Faqruddin Ali Azam (Oklahoma State University)

Suppose that \(c_k(n)\) is the average size of interval of partitions \([(0), l]\), where \(l\) runs through the set of all partitions of \(n\) with exactly \(k\) parts. We showed that the sequence of \(c_k(n)\) has polynomial growth.

Constructions of new matroids and designs over Gf(q)
Michela Ceria (University of Milan)

A perfect matroid design (PMD) is a matroid whose flats of the same rank all have the same size. In this joint work with E. Byrne, S. Ionica, R. Jurrius and E. Saçikara (arxiv:2005.03369), we introduce the q-analogue of a PMD and its properties. In order to do that, we first establish new cryptomorphic definitions for q-matroids. We show that q-Steiner systems are examples of q-PMD's and we use this matroid structure to construct subspace designs from q-Steiner systems. We apply this construction to S(2, 13, 3; q) Steiner systems and hence establish the existence of subspace designs with previously unknown parameters.

Pólya enumeration theorems in algebraic geometry
Gilyoung Cheong (University of Michigan)

Classically, the Pólya enumeration theorem concerns how to count colorings on a graph modulo symmetries. We shall see how this phenomenon also occurs in algebraic geometry.

Promotion Sorting
Colin Defant (Princeton University)

Schützenberger's promotion operator is an extensively-studied bijection that permutes the linear extensions of a finite poset. We introduce a natural extension \(\partial\) of this operator that acts on all labelings of a poset. We prove several properties of \(\partial\); in particular, we show that for every labeling \(L\) of an \(n\)-element poset \(P\), the labeling \(\partial^{n-1}(L)\) is a linear extension of \(P\). Thus, we can view the dynamical system defined by \(\partial\) as a sorting procedure that sorts labelings into linear extensions. For all \(0\leq k\leq n-1\), we characterize the \(n\)-element posets \(P\) that admit labelings that require at least \(n-k-1\) iterations of \(\partial\) in order to become linear extensions. The case in which \(k=0\) concerns labelings that require the maximum possible number of iterations in order to be sorted; we call these labelings tangled. We explicitly enumerate tangled labelings for a large class of posets that we call inflated rooted forest posets. For an arbitrary finite poset, we show how to enumerate the sortable labelings, which are the labelings \(L\) such that \(\partial(L)\) is a linear extension. This presentation is based on joint work with Noah Kravitz.

Self-dual intervals in the Bruhat order
Christian Gaetz and Yibo Gao (MIT)

Björner and Ekedahl prove that general intervals \([id, w]\) in the Bruhat order are "top-heavy", with at least as many elements in the \(i\)-th corank as the \(i\)-th rank. Well-known results of Carrell and of Lakshmibai-Sandhya give the equality case: \([id, w]\) is rank-symmetric if and only if the permutation w avoids the patterns 3412 and 4231 and these are exactly those w such that the corresponding Schubert variety is smooth. In this paper we study the finer structure of rank-symmetric intervals [id, w], beyond their rank functions. In particular, we show that these intervals are still ``top-heavy"" if one counts cover relations between different ranks. The equality case in this setting occurs when [id, w] is self-dual as a poset; we characterize these w by pattern avoidance (3412, 4231, 34521, 45321, 54123 and 54312) and in several other ways."

A Combinatorial Formula for the Kazhdan-Lusztig Polynomials of Sparse Paving Matroids
George Nasr (University of Nebraska--Lincoln)

We prove the positivity of Kazhdan-Lusztig polynomials for sparse paving matroids, which are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. The positivity follows from a remarkably simple combinatorial formula we discovered for these polynomials using skew young tableaux. This supports the conjecture that Kazhdan-Lusztig polynomials for all matroids have non-negative coeffiecients. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.

On the shifted Littlezood-Richardson coeficients and the Littlezood-Richardson coefficients
Khanh Nguyen (Institut Camille Jordan, UCBL 1, France)

We give a new interpretation of the shifted Littlewood-Richardson coefficients \(f_{\lambda\mu}^\nu\) (\(\lambda,\mu,\nu\) are strict partitions). The coefficients \(g_{\lambda\mu}\) which appear in the decomposition of Schur \(Q\)-function \(Q_\lambda\) into the sum of Schur functions \(Q_\lambda = 2^{l(\lambda)}\sum\limits_{\mu}g_{\lambda\mu}s_\mu\) can be considered as a special case of \(f_{\lambda\mu}^\nu\) (here \(\lambda\) is a strict partition of length \(l(\lambda)\)). We also give another description for \(g_{\lambda\mu}\) as the cardinal of a subset of a set that counts Littlewood-Richardson coefficients \(c_{\mu^t\mu}^{\tilde{\lambda}}\). This new point of view allows us to establish connections between \(g_{\lambda\mu}\) and \(c_{\mu^t \mu}^{\tilde{\lambda}}\). More precisely, we prove that \(g_{\lambda\mu}=g_{\lambda\mu^t}\), and \(g_{\lambda\mu} \leq c_{\mu^t\mu}^{\tilde{\lambda}}\). We conjecture that \(g_{\lambda\mu}^2 \leq c^{\tilde{\lambda}}_{\mu^t\mu}\) and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.

A crystal on decreasing factorizations in the 0-Hecke monoid
Jianping Pan & Wencin Poh (UC Davis)

We introduce a type A crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call \(\star\)-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 \(\star\)-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.

CM regularity and Kazhdan-Lusztig varieties
Colleen Robichaux (University of Illinois at Urbana-Champaign)

We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then resolve a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties using a relationship to Kazhdan-Lusztig varieties.

Complete quadrics and algebraic statistics
Tim Seynnaeve (Max Planck Institute for Mathematics in the Sciences)

Let \(L\) be a general \(d\)-dimensional linear space of symmetric \(n \times n\) matrices over \(\mathbb{C}\). What is the degree \(\phi(n,d)\) of the variety \(L^{-1}\) obtained by inverting all matrices in \(L\)? This is an interesting geometric question in its own right, but is also interesting from the point of view of algebraic statistics: \(\phi(n,d)\) is the maximum likelihood degree of the generic linear concentration model. In 2010, Sturmfels and Uhler computed \(\phi(n,d)\) for \(d \leq 5\), and conjectured that for any fixed \(d\), \(\phi(n,d)\) is a polynomial of degree \(d-1\). Using Schubert calculus and intersection theory on the space of complete quadrics, we obtain a formula for \(\phi(n,d)\) in terms of the coefficients arising in the Schur expansion of certain symmetric polynomials. Our formula allows us to prove the Sturmfels-Uhler polynomiality conjecture, and to compute the polynomials \(\phi(n,d)\) for \(d \leq 47\). This poster is based on joint work in progress with Laurent Manivel, Mateusz Michałek, Leonid Monin, Martin Vodicka, Andrzej Weber, and Jarosław Wiśniewski.

On \(b\)-symbol distances of repeated-root constacyclic codes
Tania Sidana (IIIT-Delhi, New Delhi, India)

Let \(p\) be a prime, \(s\) be a positive integer, and let \(b\) be an integer satisfying \(2 \leq b < p^s.\)   In this poster, we provide \(b\)-symbol distances of all repeated-root constacyclic codes of length \(p^s\) over finite fields. Using this result, we provide \(b\)-symbol distances of all repeated-root constacyclic codes of length \(p^s\) over finite commutative chain rings. We also list all MDS \(b\)-symbol repeated-root constacyclic codes of length \(p^s\) over finite fields, and all MDS \(b\)-symbol repeated-root constacyclic codes of length \(p^s\) over finite commutative chain rings in general.

Combinatorics of quadratic spaces over finite fields
Semin Yoo (University of Rochester)

The \(q\)-binomial coefficient \(\binom{n}{k}_{q}\) (or the Gaussian binomial coefficient) is a polynomial in \(q\), where \(q\) is a prime power. It can be described combinatorially in several ways. For example, it counts the number of \(k\)-dimensional subspaces of \(\mathbb{F}_{q}^{n}\) over \(\mathbb{F}_{q}\). In this talk, we add one more bit of structure called a quadratic form on \(\mathbb{F}_{q}^{n}\), and discuss combinatorial structures coming from quadratic spaces over finite fields in comparison with the \(q\)-binomial coefficients. This work is based on https://arxiv.org/abs/1910.03482.