Combinatorial Reciprocity for Monotone Triangles

Weak order and descents for monotone triangles

Monotone triangles are a rich extension of permutations that biject with alternating sign matrices. The notions of weak order and descent sets for permutations are generalized here to monotone triangles, and shown to enjoy many analogous properties. It is shown that any linear extension of the weak order gives rise to a shelling order on a poset, recently introduced by Terwilliger, whose maximal chains biject with monotone triangles; among these shellings are a family of EL-shellings. The weak order turns out to encode an action of the 0-Hecke monoid of type A on the monotone triangles, genera…

Refined Enumeration of Halved Monotone Triangles and Applications to Vertically Symmetric Alternating Sign Trapezoids

Halved monotone triangles are a generalisation of vertically symmetric alternating sign matrices (VSASMs). We provide a weighted enumeration of halved monotone triangles with respect to a parameter which generalises the number of $-1$s in a VSASM. Among other things, this enables us to establish a generating function for vertically symmetric alternating sign trapezoids. Our results are mainly presented in terms of constant term expressions. For the proofs, we exploit Fischer's method of operator formulae as a key tool.

Trivial Meet and Join within the Lattice of Monotone Triangles

The lattice of monotone triangles $(\mathfrak{M}_n,\le)$ ordered by entry-wise comparisons is studied. Let $τ_{\min}$ denote the unique minimal element in this lattice, and $τ_{\max}$ the unique maximum. The number of $r$-tuples of monotone triangles $(τ_1,\ldots,τ_r)$ with minimal infimum $τ_{\min}$ (maximal supremum $τ_{\max}$, resp.) is shown to asymptotically approach $r|\mathfrak{M}_n|^{r-1}$ as $n \to \infty$. Thus, with high probability this event implies that one of the $τ_i$ is $τ_{\min}$ ($τ_{\max}$, resp.). Higher-order error terms are also discussed.

Monotone triangles and 312 Pattern Avoidance

We demonstrate a natural bijection between a subclass of alternating sign matrices (ASMs) defined by a condition on the corresponding monotone triangle which we call the gapless condition and a subclass of totally symmetric self-complementary plane partitions defined by a similar condition on the corresponding fundamental domains or Magog triangles. We prove that, when restricted to permutations, this class of ASMs reduces to 312-avoiding permutations. This leads us to generalize pattern avoidance on permutations to a family of words associated to ASMs, which we call Gog words. We translate th…

弱序：交错符号矩阵、单调三角形和无碰撞管道梦想

Weak order: Alternating sign matrices, monotone triangles, and bumpless pipe dreams

本文证明了交错符号矩阵上弱序的两种定义等价，并给出了三种计算覆盖关系的方法，最后用无碰撞管道梦想刻画了弱序算子的纤维。

Refined Lattice Path Enumeration and Combinatorial Reciprocity

It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fuß-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps. We give exact formulas for the number of such paths with fixed height, fixed number of returns and (i) fixed number of valleys, (ii) fixed number of valleys with $x$-coordinate divisible by $m$ and (iii) fixed number of valleys with $x$-coordinate not divisible by $m$. The enumeration (ii) combinatorially realizes the $H$-triangle appearing in a recent article of…

Refined enumerations of alternating sign matrices: monotone (d,m)-trapezoids with prescribed top and bottom row

Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to $n \times n$ alternating sign matrices when prescribing $(1,2,...,n)$ as bottom row of the array. We define monotone $(d,m)$--trapezoids as monotone triangles with $m$ rows where the $d-1$ top rows are removed. (These objects are also equivalent to certain partial alternating sign matrices.) It is known that the number of monotone triangles with bottom row $(k_1,...,k_n)$ is given by a polynomial $α(n…

具有指定底行的单调梯形的数量

The number of monotone trapezoids with prescribed bottom row

本文建立了具有指定底行的单调梯形数量的算子公式，推广交替符号矩阵，发现特殊情况下被消去的隐藏算子，涉及自由边界并揭示与共变代数的关系，提出相关猜想。

The Zero Locus of the $F$-triangle

We are interested in the zero locus of a Chapoton's $F$-triangle as a polynomial in two real variables $x$ and $y$. An expectation is that (1) the $F$-triangle of rank $l$ as a polynomial in $x$ for each fixed $y\in[0,1]$, has exactly $l$ distinct real roots in $[0,1]$, and (2) $i$-th root $x_i(y)$ ($1 \le i \le l$) as a function on $y \in [0,1]$ is monotone decreasing. In order to understand these phenomena, we slightly generalized the concept of $F$-triangles and study the problem on the space of such generalized triangles. We analyze the case of low rank in details and show that the above e…

任意格上Jaccard距离的三角不等式研究

On the Triangle Inequality for the Jaccard Distance in Arbitrary Lattices

本文研究任意格上Jaccard距离的三角不等式，证明了满足特定条件时Jaccard距离满足三角不等式，给出必要条件，并探讨其在多计算领域的应用价值。

Dedekind-Carlitz Polynomials as Lattice-Point Enumerators in Rational Polyhedra

We study higher-dimensional analogs of the Dedekind-Carlitz polynomials c(u,v;a,b) := sum_{k=1..b-1} u^[ka/b] v^(k-1), where u and v are indeterminates and a and b are positive integers. Carlitz proved that these polynomials satisfy the reciprocity law (v-1) c(u,v;a,b) + (u-1) c(v,u;b,a) = u^(a-1) v^(b-1) - 1, from which one easily deduces many classical reciprocity theorems for the Dedekind sum and its generalizations. We illustrate that Dedekind-Carlitz polynomials appear naturally in generating functions of rational cones and use this fact to give geometric proofs of the Carlitz reciprocity…