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We prove that the symmetric function $\Delta'_{e_{k-1}}e_n$ appearing in the Delta Conjecture can be obtained from the symmetric function in the Rational Shuffle Theorem by applying a Schur skewing operator. This generalizes a formula by the first and third authors for the Delta Conjecture at $t=0$, and follows from work of Blasiak, Haiman, Morse, Pun, and Seelinger. Our main result is that we also provide a purely combinatorial proof of this skewing identity, giving a new proof of the Rise Delta Theorem from the Rational Shuffle Theorem.

We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ under the diagonal action of the symmetric group $S_n$. This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring $R_n$ due to Ariki, Terasoma, and Yamada, which has the advantage of respecting the decomposition into irreducibles. As our main application of the general theory, we provide a higher Specht basis for the hook shape Garsia--Haiman modules. In the process, we obtain a new formula for their doubly graded Frobenius series in terms of new generalized cocharge statistics on tableaux.

In this note, we give a simple closed formula for an arbitrary product, landing in dimension 0, of boundary classes on the Deligne--Mumford moduli space M_0,n-bar. For any such boundary strata $X_{T_1}, \ldots, X_{T_\ell}$, we show the intersection product $\int \prod_{i=1}^\ell [X_{T_i}]$ is either a signed product of multinomial coefficients, or zero, and provide a simple criterion for determining when it is nonzero. We do not claim originality for our product formula, but to our knowledge it does not appear elsewhere in the literature.

We prove that $\omega \Delta'_{e_{k}}e_n|_{t=0}$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $\Delta$-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Sch\"utzenberger cocharge statistic on a new combinatorial object that we call a \textit{battery-powered tableau}. Our proof is geometric, and shows that the $\Delta$-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $t$ and $t^2$ coefficients of $\omega \Delta'_{e_{k}}e_n$., Comment: Section 7 has been added, title updated

Monin and Rana conjectured a set of equations defining the image of the moduli space $\bar{M}_{0,n}$ under an embedding into $\mathbb{P}^1\times \cdots\times \mathbb{P}^{n-3}$ due to Keel and Tevelev and verified the conjecture for $n\leq 8$ using Macaulay2. We prove this conjecture for all $n$.

We consider products of $\psi$ classes and products of $\omega$ classes on $\overline{M}_{0,n+3}$. For each product, we construct a flat family of subschemes of $\overline{M}_{0,n+3}$ whose general fiber is a complete intersection representing the product, and whose special fiber is a generically reduced union of boundary strata. Our construction is built up inductively as a sequence of one-parameter degenerations, using an explicit parametrized collection of hyperplane sections. Combinatorially, our construction expresses each product as a positive, multiplicity-free sum of classes of boundary strata. These are given by a combinatorial algorithm on trees we call 'slide labeling'. As a corollary, we obtain a combinatorial formula for the $\kappa$ classes in terms of boundary strata. For degree-$n$ products of $\omega$ classes, the special fiber is a finite reduced union of (boundary) points, and its cardinality is one of the multidegrees of the corresponding embedding $\Omega_n: \overline{M}_{0,n+3}\to \mathbb{P}^1\times \cdots \times \mathbb{P}^n$. In the case of the product $\omega_1\cdots \omega_n$, these points exhibit a connection to permutation pattern avoidance. Finally, we show that in certain cases, a prior interpretation of the multidegrees via tournaments can also be obtained by degenerations.

We give a combinatorial proof of a recent geometric result of Farkas and Lian on linear series on curves with prescribed incidence conditions. The result states that the expected number of degree-$d$ morphisms from a general genus $g$, $n$-marked curve $C$ to $\mathbb{P}^r$, sending the marked points on $C$ to specified general points in $\mathbb{P}^r$, is equal to $(r+1)^g$ for sufficiently large $d$. This computation may be rephrased as an intersection problem on Grassmannians, which has a natural combinatorial interpretation in terms of Young tableaux by the classical Littlewood-Richardson rule. We give a bijection, generalizing the well-known RSK correspondence, between the tableaux in question and the $(r+1)$-ary sequences of length $g$, and we explore our bijection's combinatorial properties. We also apply similar methods to give a combinatorial interpretation and proof of the fact that, in the modified setting in which $r=1$ and several marked points map to the same point in $\mathbb{P}^1$, the number of morphisms is still $2^g$ for sufficiently large $d$.

This textbook, "Counting Rocks!", is the written component of an interactive introduction to combinatorics at the undergraduate level. Throughout the text, we link to videos where we describe the material and provide examples. The major topics in this text are counting problems (Chapters 1-4), proof techniques (Chapter 5), recurrence relations and generating functions (Chapters 6-7), and an introduction to graph theory (Chapters 8-12). The material and the problems we include are standard for an undergraduate combinatorics course. In addition to the linked videos, most chapters contain an investigation section, where students are led through a series of deeper problems on a topic. In several sections, we show students how to use the free, open source computing software SAGE in order to solve problems. We have included many illustrative figures throughout the text, and we end each section and chapter with a list of exercises of varying difficulty., Comment: 220 pages

We provide a new geometric interpretation of the multidegrees of the (iterated) Kapranov embedding $\Phi_n:\overline{M}_{0,n+3}\hookrightarrow \mathbb{P}^1\times \mathbb{P}^2\times \cdots \times \mathbb{P}^n$, where $\overline{M}_{0,n+3}$ is the moduli space of stable genus $0$ curves with $n+3$ marked points. We enumerate the multidegrees by disjoint sets of boundary points of $\overline{M}_{0,n+3}$ via a combinatorial algorithm on trivalent trees that we call a lazy tournament. These sets are compatible with the forgetting maps used to derive the recursion for the multidegrees proven in 2020 by Gillespie, Cavalieri, and Monin. The lazy tournament points are easily seen to total $(2n-1)!!=(2n-1)\cdot (2n-3) \cdots 5 \cdot 3 \cdot 1$, giving a natural proof of the fact that the total degree of $\Phi_n$ is the odd double factorial. This fact was first proven using an insertion algorithm on certain parking functions, and we additionally give a bijection to those parking functions., Comment: 22 pages, 7 figures

While equality of skew Schur functions is well understood, the problem of determining when two skew Schur $Q$ functions are equal is still largely open. It has been studied in the case of ribbon shapes in 2008 by Barekat and van Willigenburg, and this paper approaches the problem for near-ribbon shapes, formed by adding one box to a ribbon skew shape. We particularly consider frayed ribbons, that is, the near-ribbons whose shifted skew shape is not an ordinary skew shape. We conjecture, with evidence, that all Schur $Q$ functions of frayed ribbon shape are distinct up to antipodal reflection. We prove this conjecture for several infinite families of frayed ribbons, using a new approach via the "lattice walks" version of the shifted Littlewood-Richardson rule discovered in 2018 by Gillespie, Levinson, and Purbhoo., Comment: 26 pages, 12 figures. Submitted to the Electronic Journal of Combinatorics

We investigate the dynamics of the well-known RSK bijection on permutations when iterated on various reading words of the recording tableau. In the setting of the ordinary (row) reading word, we show that there is exactly one fixed point per partition shape, and that it is always reached within two steps from any starting permutation. We also consider the modified dynamical systems formed by iterating RSK on the column reading word and the reversed reading word of the recording tableau. We show that the column reading word gives similar dynamics to the row reading word. On the other hand, for the reversed reading word, we always reach either a 2-cycle or fixed point after two steps. In fact, we reach a fixed point if and only if the shape of the initial tableau is self-conjugate.

The classical coinvariant ring $R_n$ is defined as the quotient of a polynomial ring in $n$ variables by the positive-degree $S_n$-invariants. It has a known basis that respects the decomposition of $R_n$ into irreducible $S_n$-modules, consisting of the higher specht polynomials due to Ariki, Terasoma, and Yamada. We provide an extension of the higher Specht basis to the generalized coinvariant rings $R_{n,k}$. We also give a conjectured higher Specht basis for the Garsia-Procesi modules $R_\mu$, and provide a proof of the conjecture in the case of two-row partition shapes $\mu$. We then combine these results to give a higher Specht basis for an infinite subfamily of the modules $R_{n,k,\mu}$ recently defined by Griffin, which are a common generalization of $R_{n,k}$ and $R_{\mu}$., Comment: 24 pages, 4 figures

The moduli space $\overline{M}_{0,n}$ may be embedded into the product of projective spaces $\mathbb{P}^1\times \mathbb{P}^2\times \cdots \times \mathbb{P}^{n-3}$, using a combination of the Kapranov map $|\psi_n|:\overline{M}_{0,n}\to \mathbb{P}^{n-3}$ and the forgetful maps $\pi_i:\overline{M}_{0,i}\to \overline{M}_{0,i-1}$. We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height $n-3$. We use this combinatorial interpretation to show that the total degree of the embedding (thought of as the projectivization of its cone in $\mathbb{A}^2\times \mathbb{A}^3\cdots \times \mathbb{A}^{n-2}$) is equal to $(2(n-3)-1)!!=(2n-7)(2n-9) \cdots(5)(3)(1)$. As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial., Comment: 25 pages, 7 figures

We develop a combinatorial rule to compute the real geometry of type B Schubert curves $S(\lambda_\bullet)$ in the orthogonal Grassmannian $\mathrm{OG}_n$, which are one-dimensional Schubert problems defined with respect to orthogonal flags osculating the rational normal curve. Our results are natural analogs of results previously known only in type A. First, using the type B Wronski map, we show that the real locus of the Schubert curve has a natural covering map to $\mathbb{RP}^1$, with monodromy operator $\omega$ defined as the commutator of jeu de taquin rectification and promotion on skew shifted semistandard tableaux. We then introduce two different algorithms to compute $\omega$ without rectifying the skew tableau. The first uses recently-developed shifted tableau crystal operators, while the second uses local switches much like jeu de taquin. The switching algorithm further computes the K-theory coefficient of the Schubert curve: its nonadjacent switches precisely enumerate Pechenik and Yong's shifted genomic tableaux. The connection to K-theory also gives rise to a partial understanding of the complex geometry of these curves., Comment: 43 pages

We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al.. This characterization is a combination of local queer axioms generalizing Stembridge's local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph $G$ characterizing the relations of the type $A$ components of the queer crystal. We provide a counterexample to Assaf's and Oguz' conjecture that the local queer axioms uniquely characterize the queer supercrystal. We obtain a combinatorial description of the graph $G$ on the type $A$ components by providing explicit combinatorial rules for the odd queer operators on certain highest weight elements., Comment: 39 pages, 8 figures

Say that a permutation of $1,2,\ldots,n$ is \textit{$k$-bounded} if every pair of consecutive entries in the permutation differs by no more than $k$. Such a permutation is \textit{anchored} if the first entry is $1$ and the last entry is $n$. We show that the generating function for the enumeration of $k$-bounded anchored permutations is always rational, mirroring the known result on (non-anchored) $k$-bounded permutations due to Avgustinovich and Kitaev. We then explicitly determine the recursive formulas of minimal depth for the number of anchored $k$-bounded permutations of $n$ for $k=2$ and $k=3$, resolving a conjecture listed on the Online Encyclopedia of Integer Sequences (entry A249665). We additionally show that the number of anchored $k$-bounded permutations of $n$ is asymptotically $O\left(k^n\right)$ as a function of $n$ for a given $k$., Comment: This is an updated version including two new results: the rationality of the generating function for all k, and asymptotic bounds on the number of k-bounded permutations of n

We give local axioms that uniquely characterize the crystal-like structure on shifted tableaux developed in a previous paper by Gillespie, Levinson, and Purbhoo. These axioms closely resemble those developed by Stembridge for type A tableau crystals. This axiomatic characterization gives rise to a new method for proving and understanding Schur $Q$-positive expansions in symmetric function theory, just as the Stembridge axiomatic structure provides for ordinary Schur positivity.

In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear spaces. It is intended as a guide for readers with a combinatorial bent to understand and appreciate the geometric and topological aspects of Schubert calculus, and conversely for geometric-minded readers to gain familiarity with the relevant combinatorial tools in this area. We lead the reader through a tour of three variations on a theme: Grassmannians, flag varieties, and orthogonal Grassmannians. The orthogonal Grassmannian, unlike the ordinary Grassmannian and the flag variety, has not yet been addressed very often in textbooks, so this presentation may be helpful as an introduction to type B Schubert calculus. This work is adapted from the author's lecture notes for a graduate workshop during the Equivariant Combinatorics Workshop at the Center for Mathematics Research, Montreal, June 12-16, 2017., Comment: Version 2 has a typo correction

We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are the (skew) Schur $Q$-functions. We give local axioms for these crystals, which closely resemble the Stembridge axioms for type A. Finally, we give a new criterion for such tableaux to be ballot., Comment: 12 pages; Submitted to conference proceedings of Formal Power Series and Algebraic Combinatorics (FPSAC), 2018

We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but not Stembridge crystals) on the same underlying set and with the same weight functions. When taken together, the result is a new kind of `doubled crystal' structure that recovers the combinatorics of type B Schubert calculus: the highest-weight elements of our crystals are precisely the shifted Littlewood-Richardson tableaux, and their generating functions are the (skew) Schur Q functions., Comment: 35 pages, 13 included figures

We give a direct combinatorial proof of the $q,t$-symmetry relation $\tilde H_{\mu}(X;q,t)=\tilde H_{\mu'}(X;t,q)$ in the Macdonald polynomials $\tilde H_\mu$ at the specialization $q=1$. The bijection demonstrates that the Macdonald inv statistic on the permutations of any given row of a Young diagram filling is Mahonian. Moreover, our bijection gives rise a family of new bijections on words that preserves the classical Mahonian inv statistic., Comment: 10 pages; submitted to FPSAC 2017

We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(\lambda_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $\omega$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $\omega$ as the monodromy operator. We provide a local algorithm for computing $\omega$ without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the $K$-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. We then give purely combinatorial proofs of several numerical results involving the $K$-theory and real geometry of $S(\lambda_\bullet)$., Comment: 33 pages, 12 figures including 2 color figures; to appear in the Journal of Algebraic Combinatorics

We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(\lambda_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $\omega$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $\omega$ as the monodromy operator. We provide a fast, local algorithm for computing $\omega$ without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with Pechenik and Yong's genomic tableaux, which enumerate the $K$-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the $K$-theory and real geometry of $S(\lambda_\bullet)$.

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_\mu(\mathbf{x};q,t) = \widetilde{H}_{\mu^\ast}(\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $\mu$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\mathbf{x}$ for all shapes $\mu$. We also provide a proof for the full relation in the case when $\mu$ is a hook shape, and for all shapes at the specialization $t=1$. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.