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Register a new userRestricted growth function patterns and statistics
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Authors
Lindsey R. Campbell
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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.
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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.
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.
We call $i$ a fixed point of a given sequence if the value of that sequence at the $i$-th position coincides with $i$. Here, we enumerate fixed points in the class of restricted growth sequences. The counting process is conducted by calculation of generating functions and leveraging a probabilistic sampling method.
We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size $n.$ Our main results are formulas expressing the expected degree of graph nodes in terms of simple explicit functions of a finite collection of Stirling and Bernoulli numbers.
Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determine the minimum number of additional measurement devices needed to monitor a power network when the network is expanded and the existing devices remain in place. In this paper, we study the problem of finding the smallest power dominating set that contains a given set of vertices X. We also study the related problem of finding the smallest zero forcing set that contains a given set of vertices X. The sizes of such sets in a graph G are respectively called the restricted power domination number and restricted zero forcing number of G subject to X. We derive several tight bounds on the restricted power domination and zero forcing numbers of graphs, and relate them to other graph parameters. We also present exact and algorithmic results for computing the restricted power domination number, including integer programs for general graphs and a linear time algorithm for graphs with bounded treewidth. We also use restricted power domination to obtain a parallel algorithm for finding minimum power dominating sets in trees.
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