Related papers: Systematic Counting of Restricted Partitions

We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula and explicit formula. We find a connection between multidimensional permutations…

Mass partition problems describe the partitions we can induce on a family of measures or finite sets of points in Euclidean spaces by dividing the ambient space into pieces. In this survey we describe recent progress in the area in addition…

We prove a lemma that is useful to get upper bounds for the number of partitions without a given subsum. From this we can deduce an improved upper bound for the number of sets represented by the (unrestricted or into unequal parts)…

In "Square partitions and Catalan numbers" (arXiv0912.4983), Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a…

We study the enumeration of set partitions, according to their length, number of parts, cyclic type, and genus. We introduce genus-dependent Bell, Stirling numbers, and Fa\`a di Bruno coefficients. Besides attempting to summarize what is…

We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…

Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to (2+2)-free posets and…

Constrained codes are used to prevent errors from occurring in various data storage and data transmission systems. They can help in increasing the storage density of magnetic storage devices, in managing the lifetime of electronic storage…

For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for…

The number of parts in the partitions (resp. distinct partitions) of $n$ with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the…

The problem of ranking can be described as follows. We have a set of combinatorial objects $S$, such as, say, the k-subsets of n things, and we can imagine that they have been arranged in some list, say lexicographically, and we want to…

In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first $r$ elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For…

Mathematics has been used in the exploration and enumeration of juggling patterns. In the case when we catch and throw one ball at a time the number of possible juggling patterns is well-known. When we are allowed to catch and throw any…

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…

We obtain the generating functions for partial matchings avoiding neighbor alignments and for partial matchings avoiding neighbor alignments and left nestings. We show that there is a bijection between partial matchings avoiding three…

Using the slow triangle map (a type of multi-dimensional continued fraction algorithm), we exhibit a method for generating any number of new identities for subsets of integer partitions.

The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one…

The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes…

Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of…

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study…