Related papers: Set partitions with no m-nesting
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…
A classical method for partition generating functions is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with n copies of n are presented.
To flatten a set partition (with apologies to Mathematica) means to form a permutation by erasing the dividers between its blocks. Of course, the result depends on how the blocks are listed. For the usual listing--increasing entries in each…
We provide a short combinatorial proof of Cayley's formula by means of a bijective map to an outcome space of an urn-drawing problem. Furthermore we introduce an algebraic structure on the set of labeled trees, which provides a more…
The main contribution of this paper is the development of a new decision tree algorithm. The proposed approach allows users to guide the algorithm through the data partitioning process. We believe this feature has many applications but in…
Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. Many algorithms exist to generate all descending compositions, yet none have previously been published to generate…
Let $\pi$ and $\lambda$ be two set partitions with the same number of blocks. Assume $\pi$ is a partition of $[n]$. For any integer $l, m \geq 0$, let $\mathcal{T}(\pi, l)$ be the set of partitions of $[n+l]$ whose restrictions to the last…
Given $m\in \mathbb{N},$ a numerical semigroup with multiplicity $m$ is called packed numerical semigroup if its minimal generating set is included in $\{m,m+1,\ldots, 2m-1\}.$ In this work, packed numerical semigroups are used to built the…
A poset $\mathbf{P} = (X,\preceq)$ is {\em $m$-partite} if $X$ has a partition $X = X_1 \cup ... \cup X_m$ such that (1) each $X_i$ forms an antichain in $\mathbf{P}$, and (2) $x\prec y$ implies $x\in X_i$ and $y\in X_j$ where $i<j$. In…
We study the enumeration problem of higher dimensional partitions, a natural generalisation of classical integer partitions. We show that their counting problem is equivalent to the enumeration of simpler classes of higher dimensional…
We present families of combinatorial classes described as trees with nodes that can carry one of two types of "flowers": integer partitions or integer compositions. Two parameters on the flowers of trees will be considered: the number of…
Partition functions for non-interacting particles are known to be symmetric functions. It is shown that powerful group-theoretical techniques can be used not only to derive these relationships, but also to significantly simplify calculation…
We establish recursions counting various classes of chains in the noncrossing partition lattice of a finite Coxeter group. The recursions specialize a general relation which is proven uniformly (i.e. without appealing to the classification…
Let the columns of a $p \times q$ matrix $M$ over any ring be partitioned into $n$ blocks, $M = [M_1, ..., M_n]$. If no $p \times p$ submatrix of $M$ with columns from distinct blocks $M_i$ is invertible, then there is an invertible $p…
The paper introduce a new type of partitions where the largest part appears exactly once, and the remaining parts constitute a partition of that largest part. We derive the generating function associated with these partitions and…
Non-well-founded trees are used in mathematics and computer science, for modelling non-well-founded sets, as well as non-terminating processes or infinite data-structures. Categorically, they arise as final coalgebras for polynomial…
We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers-Ramanujan…
Let $c_n$ denote the number of nodes at a distance $n$ from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence $\{c_n\}$ is described. This criterion is applied…
Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…
The generating function for the number of purely crossing partitions of {1,...,n} is found in terms of the generating function for Bell numbers. Further results about generating functions for asymptotic moments of certain random Vandermonde…