Related papers: Enumeration of $n$-plexes
We give exact formulas for the number of distinct triangular patterns (or subtriangles) of a given size that occur in the Sierpi\'{n}ski Triangle.
A grid class consists of permutations whose pictorial depiction can be partitioned into increasing and decreasing parts as determined by a given matrix. In this paper, we introduce a method for enumerating cyclic permutations in vector grid…
This note will give an enumeration of $n$-cycles in the symmetric group ${\mathcal S}_n$ by their degree (also known as their cyclic descent number) and studies similar counting problems for the conjugacy classes of $n$-cycles under the…
P(n,s) denotes the number of permutations of 1,2,...n that have exactly s sequences. Canfield and Wilf [math.CO/0609704] recently showed that P(n,s) can be written as a sum of s polynomials in n. We determine these polynomials explicitly…
The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary…
We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.
In recent work, Zeilberger and the author used a functional equations approach for enumerating permutations with r occurrences of the pattern 12...k. In particular, the approach yielded a polynomial-time enumeration algorithm for any fixed…
We consider the problem of enumerating periodic $\sigma$-juggling sequences of length $n$ for multiplex juggling, where $\sigma$ is the initial state (or {\em landing schedule}) of the balls. We first show that this problem is equivalent to…
In 2021, Gupta and Suzumura proposed a novel algorithm for enumerating all bounded-length simple cycles in directed graphs. In this work, we present concrete examples demonstrating that the proposed algorithm fails to enumerate certain…
Biclustering involves the simultaneous clustering of objects and their attributes, thus defining local two-way clustering models. Recently, efficient algorithms were conceived to enumerate all biclusters in real-valued datasets. In this…
Cyclic codes have attracted a lot of research interest for decades. In this paper, for an odd prime $p$, we propose a general strategy to compute the complete weight enumerator of cyclic codes via the value distribution of the corresponding…
We find exact and asymptotic formulas for the number of pairs $(p,q)$ of $N$-cycles such that the all cycles of the product $p\cdot q$ have lengths from a given integer set. We then apply these results to prove a surprisingly high lower…
We prove an asymptotic formula for the number of Eulerian circuits for graphs with strong mixing properties and with vertices having even degrees. The exact value is determined up to the multiplicative error $O(n^{-1/2+\varepsilon})$, where…
We develop general formulae for the numbers of conjugacy classes and irreducible complex characters of finite p-groups of nilpotency class less than p. This allows us to unify and generalize a number of existing enumerative results, and to…
We enumerate total cyclic orders on $\left\{1,\ldots,n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(i,{i+1},{i+2})$, these integers being taken modulo $n$. In some cases, the problem reduces to the…
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…
We review the methods to combine several measurements, in the form of parameter values or $p$-values.
Polytope numbers for a given polytope are an integer sequence defined by the combinatorics of the polytope. Recent work by H. K. Kim and J. Y. Lee has focused on writing polytope number sequences as sums of simplex number sequences. We…
In this paper, we describe linear and cyclic codes over the rings of the form $R_{s,p}=\mathbb{Z}_{p}[u]/\left( f\left(u\right) /\left( u-s\right) \right)$, where $p$ is a prime number and $f\left( u\right) =u^{p}-u$, with $s\in…
We address the problem of enumerating paths in square lattices, where allowed steps include (1,0) and (0,1) everywhere, and (1,1) above the diagonal y=x. We consider two such lattices differing in whether the (1,1) steps are allowed along…