Linear extensions of posets are important objects in enumerative and algebraic combinatorics that are difficult to count in general. Families of posets like Young diagrams of straight shapes and $d$-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula from 2014. In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine $d$-complete posets and border strip skew shapes. We give a Naruse type hook-length formula to count linear extensions of such posets by proving a major index $q$-analogue. We also a inversion index $q$-analogue of the Naruse formula for mobile tree posets.
The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In 2015 we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations.
We present a conjectual hook formula concerning the number of the standard tableaux on "cylindric" skew diagrams. Our formula can be seen as an extension of Naruse's hook formula for skew diagrams. Moreover, we prove our conjecture in some special cases.
We construct a hook-content formula and its $q$-analog using excited Young diagrams analogous to Naruse's hook-length formula for skew shapes. Furthermore, we show that our hook-content formula has a simple factorization and give some conjectures and questions related to its $q$-analog.
The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman--Grassl correspondence, respectively. The main new results are two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting rever…
We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes our formulas generalize the classical hook-length formula and Stanley's formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its $q$-analogues, which were studied in previous papers of the series.
Peterson and Proctor obtained a formula which expresses the multivariate generating function for $P$-partitions on a $d$-complete poset $P$ as a product in terms of hooks in $P$. In this paper, we give a skew generalization of Peterson--Proctor's hook formula, i.e., a formula for the generating function for $(P \setminus F)$-partitions for a $d$-complete poset $P$ and its order filter $F$, by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant $K$-theory of Kac--Moody partial flag varieties. This generalization provides…
We introduce a class of posets, which includes both ribbon posets (skew shapes) and $d$-complete posets, such that their number of linear extensions is given by a determinant of a matrix whose entries are products of hook lengths. We also give $q$-analogues of this determinantal formula in terms of the major index and inversion statistics. As applications, we give families of tree posets whose numbers of linear extensions are given by generalizations of Euler numbers, we draw relations to Naruse-Okada's positive formulas for the number of linear extensions of skew $d$-complete posets, and we g…
We define a new natural partial order on Motzkin paths that serves as an intermediate step between two previously-studied partial orders. We provide a bijection between valid hook configurations of $312$-avoiding permutations and intervals in these new posets. We also show that valid hook configurations of permutations avoiding $132$ (or equivalently, $231$) are counted by the same numbers that count intervals in the Motzkin-Tamari posets that Fang recently introduced, and we give an asymptotic formula for these numbers. We then proceed to enumerate valid hook configurations of permutations av…
The hook length formula for $d$-complete posets states that the $P$-partition generating function for them is given by a product in terms of hook lengths. We give a new proof of the hook length formula using $q$-integrals. The proof is done by a case-by-case analysis consisting of two steps. First, we express the $P$-partition generating function for each case as a $q$-integral and then we evaluate the $q$-integrals. Several $q$-integrals are evaluated using partial fraction expansion identities and others are verified by computer.
使用一个修改版本的jeu de taquin，Novelli，Pak和Stoyanovskii给出了一个双射证明的钩长公式计数标准杨tableaux的固定形状。本文考虑Jeu de Taquin到任意偏序集的一个自然扩张。给定一个偏序集P，jeu de taquin定义了一个从偏序集元素的双射标号集到偏序集的线性扩张集的映射，该偏序集元素的双射标号集具有$\{1,2,...,|P|\}$。一个特别有趣的问题是，对于哪些偏序集，这个映射产生每个线性扩张的频率相等。我们分析了双尾钻石偏序集$D_{m,n}$，证明了均匀分布的存在性当且仅当$D_{m,n}$是d-完全的.此外，我们观察到，扩展的钩长公式计数线性扩展的d-完全偏序集提供了一个组合的答案，一个看似无关的问题，即：给定一个均匀随机的标准杨表的固定形状，什么是期望值的最左边的条目在第二行？
d-Complete posets: local structural axioms, properties, and equivalent definitions
虽然d-完备偏序集是沿着代数组合学和李理论之间的界面出现的，但它们的定义只需要它们的局部结构。这些偏序集是有根树、形状和移位形状的相互推广。他们拥有斯坦利的钩产品的P-分区生成功能和Schutzenberger的定义良好的jeu de taquin整流财产财产。d-完全偏序集的最初定义很长，但后来发展出更简洁的定义。这里有几个定义是等价的。总结了d-完备偏序集的基本性质。背景和这些偏序集的部分书目。
