Related papers: Twisted Schubert polynomials
We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $\mathcal{P}(\pi)$ denote the set of indices where $\pi$ has a peak. Given a set $S$ of positive…
A beautiful degree formula for the Grothendieck polynomials was recently given by Pechenik, Speyer, and Weigandt (2021). We provide an alternative proof of their degree formula, utilizing the climbing chain model for Grothendieck…
Motivated by Stanley's ``Schubert shenanigans'' paper, commendable attempts have been made to understand the principal specializations of Schubert or Grothendieck polynomials. In this paper, we prove that when a permutation $w$ does not…
For a permutation $u\in S_n$, let $N\ast u\in S_{Nn}$ be the permutation with scaled Lehmer code. For given $u,v,w\in S_n$ and integer $N$, the stretched Schubert coefficients are defined as $f_{u,v,w}(N):=c_{N*u,N*v}^{N*w}$. Our main…
The M-convexity of dual Schubert polynomials was first proven by Huh, Matherne, M\'esz\'aros, and St. Dizier in 2022. We give a full characterization of the supports of dual Schubert polynomials, which yields an elementary alternative proof…
We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown…
In the present work, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of new monogenic polynomials are provided based on 2-parameters weight functions. Such classes extend the well…
Let $N_\sigma(\pi)$ denote the number of occurrences of a permutation pattern $\sigma\in S_k$ in a permutation $\pi\in S_n$. Gaetz and Ryba (2021) showed using partition algebras that the $d$-th moment $M_{\sigma,d,n}(\pi)$ of $N_\sigma$ on…
Some new relations on skew Schur function differences are established both combinatorially using Sch\"utzenberger's jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain…
Real algebraic geometry provides certificates for the positivity of polynomials on semi-algebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalitites. We show how Putinar's theorem for…
Schubert polynomials are refined by the key polynomials of Lascoux-Sch\"{u}tzenberger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles. In this paper we determine which fundamental slide polynomial…
In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it.
Denote by $u(n)$ the largest principal specialization of the Schubert polynomial: $ u(n) := \max_{w \in S_n} \mathfrak{S}_w(1,\ldots,1) $ Stanley conjectured in [arXiv:1704.00851] that there is a limit $\lim_{n\to \infty} \, \frac{1}{n^2}…
We prove Pieri formulas for the multiplication with special Schubert classes in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian…
The coefficients of the Kazhdan-Lusztig polynomials $P_{v,w}(q)$ are nonnegative integers that are upper semicontinuous on Bruhat order. Conjecturally, the same properties hold for $h$-polynomials $H_{v,w}(q)$ of local rings of Schubert…
We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products Grothendieck polynomials in each set of…
We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian…
We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule…
We investigate the longstanding problem of finding a combinatorial rule for the Schubert structure constants in the $K$-theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux-M.-P. Sch\"{u}tzenberger (1982) serve…
We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic…