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The $k$-arrangements are permutations whose fixed points are $k$-colored. We prove enumerative results related to statistics and patterns on $k$-arrangements, confirming several conjectures by Blitvic and Steingrimsson. In particular, one of their conjectures regarding the equdistribution of the number of descents over the derangement form and the permutation form of $k$-arrangements is strengthened in two interesting ways. Moreover, as one application of the so-called Decrease Value Theorem, we calculate the generating function for a symmetric pair of Eulerian statistics over permutations arising in our study.
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A set partition $sigma$ of $[n]={1,dots,n}$ contains another set partition $pi$ if restricting $sigma$ to some $Ssubseteq[n]$ and then standardizing the result gives $pi$. Otherwise we say $sigma$ avoids $pi$. For all sets of patterns consisting of partitions of $[3]$, the sizes of the avoidance classes were determined by Sagan and by Goyt. Set partitions are in bijection with restricted growth functions (RGFs) for which Wachs and White defined four fundamental statistics. We consider the distributions of these statistics over various avoidance classes, thus obtaining multivariate analogues of the previously cited cardinality results. This is the first in-depth study of such distributions. We end with a list of open problems.
A restricted growth function (RGF) of length n is a sequence w = w_1 w_2 ... w_n of positive integers such that w_1 = 1 and w_i is at most 1 + max{w_1,..., w_{i-1}} for i at least 2. RGFs are of interest because they are in natural bijection with set partitions of {1, 2, ..., n}. RGF w avoids RGF v if there is no subword of w which standardizes to v. We study the generating functions sum_{w in R_n(v)} q^{st(w)} where R_n(v) is the set of RGFs of length n which avoid v and st(w) is any of the four fundamental statistics on RGFs defined by Wachs and White. These generating functions exhibit interesting connections with integer partitions and two-colored Motzkin paths, as well as noncrossing and nonnesting set partitions.
In this article we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual-partition formula. Then it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl arrangement such that each filter is a free subarrangement satisfying the dual-partition formula. This generalizes the main result in cite{ABCHT} which affirmatively settled a conjecture by Sommers and Tymoczko cite{SomTym}.
The Laplacian matrix of a graph G describes the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G. The lattice ideal associated to the lattice generated by the columns of the Laplacian provides an algebraic perspective on this recently (re)emerging field. This ideal I_G has a distinguished monomial initial ideal M_G, characterized by the property that the standard monomials are in bijection with the G-parking functions of the graph G. The ideal M_G was also introduced by Postnikov and Shapiro (2004) in the context of monotone monomial ideals. We study resolutions of M_G and show that a minimal free cellular resolution is supported on the bounded subcomplex of a section of the graphical arrangement of G. This generalizes constructions from Postnikov and Shapiro (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and of Perkinson et al. on the commutative algebra of Sandpiles. As a corollary we verify a conjecture of Perkinson et al. regarding the Betti numbers of M_G, and in the process provide a combinatorial characterization in terms of acyclic orientations.
In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements.
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