Monotone triangles are certain triangular arrays of integers, which correspond to $n \times n$ alternating sign matrices when prescribing $(1,2,...,n)$ as bottom row of the monotone triangle. In this article we define halved monotone triangles, a specialization of which correspond to vertically symmetric alternating sign matrices. We derive an operator formula for the number of halved monotone triangles with prescribed bottom row which is analogous to our operator formula for the number of ordinary monotone triangle.
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.
We show that the number of monotone triangles with prescribed bottom row (k_1,...,k_n) is given by a simple product formula which remarkably involves (shift) operators. Monotone triangles with bottom row (1,2,...,n) are in bijection with $n \times n$ alternating sign matrices.
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…
We provide a simplified proof of our operator formula for the number of monotone triangles with prescribed bottom row, which enables us to deduce three generalizations of the formula. One of the generalizations concerns a certain weighted enumeration of monotone triangles which specializes to the weighted enumeration of alternating sign matrices with respect to the number of -1s in the matrix when prescribing $(1,2,...,n)$ as the bottom row of the monotone triangle.
In a recent work, the combinatorial interpretation of the polynomial alpha(n;k1,k2,...,kn) counting the number of Monotone Triangles with bottom row k1 < k2 < ... < kn was extended to weakly decreasing sequences k1 >= k2 >= ... >= kn. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles - a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of alpha(n;k1,k2,...,kn) at a…
The number of Monotone Triangles with bottom row k1 < k2 < ... < kn is given by a polynomial alpha(n; k1,...,kn) in n variables. The evaluation of this polynomial at weakly decreasing sequences k1 >= k2 >= ... >= kn turns out to be interpretable as signed enumeration of new combinatorial objects called Decreasing Monotone Triangles. There exist surprising connections between the two classes of objects -- in particular it is shown that alpha(n; 1,2,...,n) = alpha(2n; n,n,n-1,n-1,...,1,1). In perfect analogy to the correspondence between Monotone Triangles and Alternating Sign Matrices, the set …
The {\em line sum optimization problem} asks for a $(0,1)$-matrix minimizing the sum of given functions evaluated at its row and column sums. We show that the {\em uniform} problem, with identical row functions and identical column functions, and the {\em monotone} problem, over matrices with nonincreasing row and column sums, are polynomial time solvable.
We consider ribbon shapes, not necessarily connected, whose rows, with at least two boxes in each, are in monotone length order. These ribbons are uniquely defined by a pair of partitions: the row partition consisting of the row lengths in decreasing order, and the overlapping partition whose entries count the total number of columns with two boxes in the successive ribbon shapes obtained by sequentially subtracting the longest row. The support of such ribbon Schur functions, considered as a subposet of the dominance order lattice on partitions, has the row partition as bottom element, and, as…
The weighted projection of an alternating sign matrix (ASM) was introduced by Brualdi and Dahl (2018) as a step towards characterising a generalisation of Latin squares they introduced using alternating sign hypermatrices. If $z_n = (n,\dots,2,1)$, then the weighted projection of an ASM $A$ is equal to $z_n^TA$. Brualdi and Dahl proved that the weighted projection of an $n \times n$ ASM is majorized by the vector $z_n$, and conjectured that any positive integer vector majorized by $z_n$ is the weighted projection of some ASM. The main result of this paper presents a proof of this conjecture, v…
In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of $n \times n$ alternating sign matrices with a 1 at the top of the $i$-th column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0-1-(-1) matrices generalizing alternating sign matrices.
We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions are, depending on the number of fixed points, (1) the Tracy-Widom distributions for the largest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy-Widom distributions. We also consider the second rows of the correspond…
Alternating sign triangles (ASTs) have recently been introduced by Ayyer, Behrend and the author, and it was proven that there is the same number of ASTs with n rows as there is of nxn alternating sign matrices (ASMs). We prove a conjecture by Behrend on a refined enumeration of ASTs with respect to a statistic that is shown to have the same distribution as the column of the unique 1 in the top row of an ASM. The proof of the conjecture is based on a certain multivariate generating function of ASTs that takes the positions of the columns with sum 1 (1-columns) into account. We also prove a cur…
Gog and Magog trapezoids are certain arrays of positive integers that generalize alternating sign matrices (ASMs) and totally symmetric self-complementary plane partitions (TSSCPPs) respectively. Zeilberger used constant term formulas to prove that there is the same number of (n,k)-Gog trapezoids as there is of (n,k)-Magog trapezoids, thereby providing so far the only proof for a weak version of a conjecture by Mills, Robbins and Rumsey from 1986. About 20 years ago, Krattenthaler generalized Gog and Magog trapezoids and formulated an extension of their conjecture, and, recently, Biane and Che…
We show that there is the same number of (n,l)-alternating sign trapezoids as there is of column strict shifted plane partitions of class l-1 with at most n parts in the top row, thereby proving a result that was conjectured independently by Behrend and Aigner. The first objects generalize alternating sign triangles, which have recently been introduced by Ayyer, Behrend and the author who showed that they are counted by the same formula as alternating sign matrices. Column strict shifted plane partitions of a fixed class were introduced in a slightly different form by Andrews, and they essenti…
There is the same number of $n \times n$ alternating sign matrices (ASMs) as there is of descending plane partitions (DPPs) with parts no greater than $n$, but finding an explicit bijection is an open problem for about $40$ years now. So far, quadruples of statistics on ASMs and on DPPs that have the same joint distribution have been identified. We introduce extensions of ASMs and of DPPs along with $n+3$ statistics on each extension, and show that the two families of statistics have the same joint distribution. The ASM-DPP equinumerosity is obtained as an easy consequence by considering the $…
