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We consider the (iterated) Kapranov embedding \(\Omega_n:\overline{M}_{0,n+3} \hookrightarrow \mathbb{P}^1 \times \cdots \times \mathbb{P}^n\), where \(\overline{M}_{0,n+3}\) is the moduli space of stable genus \(0\) curves with \(n+3\) marked points. In 2020, Gillespie, Cavalieri, and Monin gave a recursion satisfied by the multidegrees of \(\Omega_n\) and showed, using two combinatorial insertion algorithms on certain parking functions, that the total degree of \(\Omega_n\) is \((2n-1)!!=(2n-1)\cdot (2n-3) \cdots 5 \cdot 3 \cdot 1\). In this paper, we give a new proof of this fact by enumerating each multidegree by a set of boundary points of \(\overline{M}_{0,n+3}\), via an algorithm on trivalent trees that we call a lazy tournament. The advantages of this new interpretation are twofold: first, these sets project to one another under the forgetting maps used to derive the multidegree recursion. Second, these sets naturally partition the complete set of boundary points on \(\overline{M}_{0,n+2}\), of which there are \((2n-1)!!\), giving an immediate proof of the total degree formula.Mathematics Subject Classifications: 05E14, 14N10, 05C05, 14H10, 05A19, 05C85Keywords: Moduli spaces of curves, projective embeddings, multidegrees, trivalent trees

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 supercrystal. 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. This also yields a new combinatorial description of the Schur expansion of the Schur P-polynomials.

We prove that |$\omega \Delta ^{\prime}_{e_{k}}e_{n}|_{t=0}$|⁠ , 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 $| -Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered tableau. Our proof is geometric, and shows that the |$\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 ^{\prime}_{e_{k}}e_{n}$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we consider products of ψ and ω classes on M ¯ 0 , n + 3 . For each product, we construct a flat family of subschemes of 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 positive combinatorial formula for the κ classes in terms of boundary strata. For degree-n products of ω classes, the special fiber is a finite reduced union of (boundary) points, and its cardinality is one of the multidegrees of the corresponding embedding Ω n : M ¯ 0 , n + 3 → P 1 × ⋯ × P n . In the case of the product ω 1 ⋯ ω 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. [ABSTRACT FROM AUTHOR]

Controversy surrounds the use of standardized exit exams to predict and prepare students for NCLEX success. The purpose of this study was to explore the relationships between Kaplan integrated exam scores and the HESI® exit exam score to determine early indicators of success on the exit exam. A retrospective records review was used to explore relationships between performance on integrated tests and the standardized exit exam. A solid Kaplan comprehensive medical surgical exam score was highly correlated with success on the HESI exit exam. Tracking and follow-up resulted in exit exam success for nearly all students. [ABSTRACT FROM AUTHOR]