Initiated by Davis, Nelson, Petersen and Tenner (2018), the enumerative study of pinnacle sets of permutations has attracted a fair amount of attention recently. In this article, we provide a recurrence that can be used to compute efficiently the number $|\mathfrak{S}_n(P)|$ of permutations of size $n$ with a given pinnacle set $P$, with arithmetic complexity $O(k^4 + k\log n)$ for $P$ of size $k$. A symbolic expression can also be computed in this way for pinnacle sets of fixed size. A weighted sum $q_n(P)$ of $|\mathfrak{S}_n(P)|$ proposed in Davis, Nelson, Petersen and Tenner (2018) seems to have a simple form, and a conjectural form is given recently by Flaque, Novelli and Thibon (2021+). We settle the problem by providing and proving an alternative form of $q_n(P)$, which has a strong combinatorial flavor. We also study admissible orderings of a given pinnacle set, first considered by Rusu (2020) and characterized by Rusu and Tenner (2021), and we give an efficient algorithm for their counting.
In previous papers, for an arithmetical function $f_0$, we defined functions $f_m$ and $c_m$ and designated numbers of restricted words over a finite alphabet counted by these functions. In this paper, we examine the reverse problem for five specific types of restricted words. Namely, we find the initial function $f_0$ such that $f_m$ and $c_m$ enumerate these words. In each case, we derive explicit formulas for $f_m$ and $c_m$. Fibonacci, Merssen, Pell, Jacosthal, Tribonacci, and Padovan numbers all appear as values of $f_m$, so we obtain new formulas for these numbers. Also, we combinatorial…
A multipermutation with $k$ copies each of $1\ldots n$ is Carlitz if neighbours are different. We enumerate these objects for $k=2,3,4$ and derive recurrences. In particular, we prove and improve a conjectured recurrence for $k=3$, stated in OEIS, the Online Encyclopedia of Integer Sequences.
In this paper, we count a dual set of Stirling permutations by the number of alternating runs. Properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas are studied.
Regular structures of an intractable enumeration problem: a diagonal recurrence relation of monomer-polymer coverings on two-dimensional rectangular lattices
For enumerative problems, i.e. computable functions f from N to Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f(n) in the number of steps that is polynomial in the combined size of the input n and the output f(n), both written in binary notation. We discuss many examples of enumerative problems for which such closed formulas are, or are not, known. These problems include (i) linear recurrence sequences and holonomic sequences, (ii) integer partitions, (iii) pattern-avoiding permutations, (iv) triangle-free graphs and (v) regular graphs. In part I we …
In this work, we generalize a recursive enumerative formula for connected Feynman diagrams with two external legs. The Feynman diagrams are defined from a fermionic gas with a two-body interaction. The generalized recurrence is valid for connected Feynman diagrams with an arbitrary number of external legs and an arbitrary order. The recurrence formula terms are expressed in function of weak compositions of non-negative integers and partitions of positive integers in such a way that to each term of the recurrence correspond a partition and a weak composition. The foundation of this enumeration …
We describe a class of fixed polyominoes called $k$-omino towers that are created by stacking rectangular blocks of size $k\times 1$ on a convex base composed of these same $k$-omino blocks. By applying a partition to the set of $k$-omino towers of fixed area $kn$, we give a recurrence on the $k$-omino towers therefore showing the set of $k$-omino towers is enumerated by a Gauss hypergeometric function. The proof in this case implies a more general hypergeometric identity with parameters similar to those given in a classical result of Kummer.
In this paper, we find an explicit formula for the generating function that counts the circular permutations of length n avoiding the pattern 23 4 1 whose enumeration was raised as an open problem by Rupert Li. This then completes in all cases the enumeration of circular permutations that avoid a single vincular pattern of length four containing one vinculum. To establish our results, we introduce three auxiliary arrays which when taken together refine the cardinality of the class of permutations in question. Rewriting the recurrences of these arrays in terms of generating functions leads to f…
We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions $λ$ whose Young diagrams fit in an $m$ by $n$ box. We show that these polynomials can be expressed by the finite difference operator applied to the power $x^{m+n}$. Evaluating box polynomials yields a variety of identities involving set partition enumeration. We extend the latter identities using restricted growth words and a new operator called the fast Fourier operator, and consider connections between set partition enumeration and the chromatic polynomial on graphs. We also give conn…
The aim of this paper is to develop the combinatorics of constructions associated to what we call \emph{triangular partitions}. As introduced in arXiv:2102.07931, these are the partitions whose cells are those lying below the line joining points $(r,0)$ and $(0,s)$, for any given positive reals $r$ and $s$. Classical notions such as Dyck paths and parking functions are naturally generalized by considering the set of partitions included in a given triangular partition. One of our striking results is that the restriction of the Young lattice to triangular partition has a planar Hasse diagram, wi…
We study set partitions with $r$ distinguished elements and block sizes found in an arbitrary index set $S$. The enumeration of these $(S,r)$-partitions leads to the introduction of $(S,r)$-Stirling numbers, an extremely wide-ranging generalization of the classical Stirling numbers and the $r$-Stirling numbers. We also introduce the associated $(S,r)$-Bell and $(S,r)$-factorial numbers. We study fundamental aspects of these numbers, including recurrence relations and determinantal expressions. For $S$ with some extra structure, we show that the inverse of the $(S,r)$-Stirling matrix encodes th…
Given a digraph $G$, we propose a new method to find the recurrence equation for the number of vertices $n_k$ of the $k$-iterated line digraph $L^k(G)$, for $k\geq0$, where $L^0(G)=G$. We obtain this result by using the minimal polynomial of a quotient digraph $π(G)$ of $G$. We show some examples of this method applied to the so-called cyclic Kautz, the unicyclic, and the acyclic digraphs. In the first case, our method gives the enumeration of the ternary length-2 squarefree words of any length.
Normal forms for wide classes of closed IL formulas were given in [4]. Here we quantify asymptotically, in exact numbers, how wide those classes are. As a consequence, we show that the "majority" of closed IL formulas have GL-equivalents, and by that, they have the same normal forms as GL formulas. Our approach is entirely syntactical, except for applying the results of [4]. As a byproduct we devise a convenient way of computing asymptotic behaviors of somewhat general classes of formulas given by their grammar rules. Its applications do not require any knowledge of the recurrence relations, g…
