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Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo–Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo–Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo–Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo–Mumford polynomials agree up to scalar multiple. The degree of the Grothendieck polynomial is a new permutation statistic which we call the Rajchgot index; we develop the properties of Rajchgot index and relate it to major index and to weak order. 

Abstract We introduce two new bases of the ring of polynomials and study their relations to known bases. The first basis is the quasi-Lascoux basis, which is simultaneously both a $K$ -theoretic deformation of the quasi-key basis and also a lift of the $K$ -analogue of the quasi-Schur basis from quasi-symmetric polynomials to general polynomials. We give positive expansions of this quasi-Lascoux basis into the glide and Lascoux atom bases, as well as a positive expansion of the Lascoux basis into the quasi-Lascoux basis. As a special case, these expansions give the first proof that the $K$ -analogues of quasi-Schur polynomials expand positively in multifundamental quasi-symmetric polynomials of T. Lam and P. Pylyavskyy. The second new basis is the kaon basis, a $K$ -theoretic deformation of the fundamental particle basis. We give positive expansions of the glide and Lascoux atom bases into this kaon basis. Throughout, we explore how the relationships among these $K$ -analogues mirror the relationships among their cohomological counterparts. We make several “alternating sum” conjectures that are suggestive of Euler characteristic calculations. 

Abstract One of the oldest outstanding problems in dynamical algebraic combinatorics is the following conjecture of P. Cameron and D. Fon-Der-Flaass (1995): consider a plane partition P in an $a \times b \times c$ box ${\sf B}$ . Let $\Psi (P)$ denote the smallest plane partition containing the minimal elements of ${\sf B} - P$ . Then if $p= a+b+c-1$ is prime, Cameron and Fon-Der-Flaass conjectured that the cardinality of the $\Psi $ -orbit of P is always a multiple of p . This conjecture was established for $p \gg 0$ by Cameron and Fon-Der-Flaass (1995) and for slightly smaller values of p in work of K. Dilks, J. Striker and the second author (2017). Our main theorem specializes to prove this conjecture in full generality.