相关论文: Hook length property of $d$-complete posets via $q…

We define semi-pointed partition posets, which are a generalisation of partition posets and show that they are Cohen-Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on…

We define the notion of connectivity set for elements of any finitely generated Coxeter group. Then we define an order related to this new statistic and show that the poset is graded and each interval is a shellable lattice. This implies…

Using the theory of functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, then, it can be expanded in terms of the product of the…

The combinatorial properties of partitions with various restrictions on their hooksets are explored. A connection with numerical semigroups extends current results on simultaneous s/t-cores. Conditions that suffice for a partition to…

We present a method of computing the generating function $f_P(\x)$ of $P$-partitions of a poset $P$. The idea is to introduce two kinds of transformations on posets and compute $f_P(\x)$ by recursively applying these transformations. As an…

For a simple connected graph $G$, the $Q$-generating function of the numbers $N_k$ of semi-edge walks of length $k$ in $G$ is defined by $W_Q(t)=\sum\nolimits_{k = 0}^\infty {N_k t^k }$. This paper reveals that the $Q$-generating function…

For Coxeter groups with sufficiently large braid relations, we prove that the sequence of powers of a Coxeter element has unbounded reflection length. We establish a connection between the reflection length functions on arbitrary Coxeter…

The classical 1991 result by Brightwell and Winkler states that the number of linear extensions of a poset is #P-complete. We extend this result to posets with certain restrictions. First, we prove that the number of linear extension for…

A factorization of an element $x$ in a monoid $(M, \cdot)$ is an expression of the form $x = u_1^{z_1} \cdots u_k^{z_k}$ for irreducible elements $u_1, \ldots, u_k \in M$, and the length of such a factorization is $z_1 + \cdots + z_k$. We…

In a recent paper, Bringmann, Craig, Ono, and the author showed that the number of $t$-hooks ($t\geq2$) among all partitions of $n$ is not always asymptotically equidistributed on congruence classes $a \pmod{b}$. In this short note, we…

We establish a hook length bias between self-conjugate partitions and partitions of distinct odd parts, demonstrating that there are more hooks of fixed length $t \geq 2$ among self-conjugate partitions of $n$ than among partitions of…

Folding subgroups give a way to realize non-simply-laced Coxeter groups as subgroups of simply-laced Coxeter groups. In this paper, we study how folding subgroups of finite and affine type are distributed length-wise by calculating the…

The set of hook lengths of an integer partition $\lambda$ is the complement of some numerical semigroup $S$. There has been recent interest in studying the number of partitions with a given set of hook lengths. Very little is known about…

We consider sequences of integers defined by a system of linear inequalities with integer coefficients. We show that when the constraints are strong enough to guarantee that all the entries are nonnegative, the generating function for the…

Recently, a new weighted generalization of the branching rule for the hook lengths, equivalent to the hook formula, was proved. In this paper, we generalize the complementary branching rule, which can be used to prove Burnside's formula. We…

The Nekrasov--Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The…

The Macdonald polynomials can be obtained by acting on the constant 1 with creation operators. Three different expressions for these operators are derived, one from the other, in a rather succint way. When the last of these expressions is…

We introduce stable reflection length in Coxeter groups, as a way to study the asymptotic behaviour of reflection length. This creates connections to other well-studied stable length functions in groups, namely stable commutator length and…

Let $G$ be a finite soluble group and $h(G)$ its Fitting length. The aim of this paper is to give certain upper bounds for $h(G)$ as functions of the Fitting length of at least three Hall subgroups of $G$ which factorize $G$ in a particular…

In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is…