17,574 research outputs found
On the maximal weight of -ary chain partitions with bounded parts
A -ary chain is a special type of chain partition of integers with
parts of the form for some fixed integers and . In this note,
we are interested in the maximal weight of such partitions when their parts are
distinct and cannot exceed a given bound . Characterizing the cases where
the greedy choice fails, we prove that this maximal weight is, as a function of
, asymptotically independent of , and we provide an efficient
algorithm to compute it.Comment: 17 page
A Graph-theoretic Method to Define any Boolean Operation on Partitions
The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century. Yet there is a simple and natural graph-theoretic method presented here to define any n-ary Boolean operation on partitions. An equivalent closure-theoretic method is also defined. In closing, the question is addressed of why it took so long for all Boolean operations to be defined for partitions
Enumerative Coding for Grassmannian Space
The Grassmannian space \Gr is the set of all dimensional subspaces of
the vector space~\smash{\F_q^n}. Recently, codes in the Grassmannian have
found an application in network coding. The main goal of this paper is to
present efficient enumerative encoding and decoding techniques for the
Grassmannian. These coding techniques are based on two different orders for the
Grassmannian induced by different representations of -dimensional subspaces
of \F_q^n. One enumerative coding method is based on a Ferrers diagram
representation and on an order for \Gr based on this representation. The
complexity of this enumerative coding is digit
operations. Another order of the Grassmannian is based on a combination of an
identifying vector and a reduced row echelon form representation of subspaces.
The complexity of the enumerative coding, based on this order, is
digits operations. A combination of the two
methods reduces the complexity on average by a constant factor.Comment: to appear in IEEE Transactions on Information Theor
On Non-Squashing Partitions
A partition n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k is
called non-squashing if p_1 + ... + p_j <= p_{j+1} for 1 <= j <= k-1.
Hirschhorn and Sellers showed that the number of non-squashing partitions of n
is equal to the number of binary partitions of n. Here we exhibit an explicit
bijection between the two families, and determine the number of non-squashing
partitions with distinct parts, with a specified number of parts, or with a
specified maximal part. We use the results to solve a certain box-stacking
problem.Comment: 15 pages, 2 fig
Generalized Connectives for Multiplicative Linear Logic
In this paper we investigate the notion of generalized connective for multiplicative linear logic. We introduce a notion of orthogonality for partitions of a finite set and we study the family of connectives which can be described by two orthogonal sets of partitions.
We prove that there is a special class of connectives that can never be decomposed by means of the multiplicative conjunction ? and disjunction ?, providing an infinite family of non-decomposable connectives, called Girard connectives. We show that each Girard connective can be naturally described by a type (a set of partitions equal to its double-orthogonal) and its orthogonal type. In addition, one of these two types is the union of the types associated to a family of MLL-formulas in disjunctive normal form, and these formulas only differ for the cyclic permutations of their atoms
Combinatorially interpreting generalized Stirling numbers
Let be a word in alphabet with 's and 's.
Interpreting "" as multiplication by , and "" as differentiation with
respect to , the identity , valid
for any smooth function , defines a sequence , the terms of
which we refer to as the {\em Stirling numbers (of the second kind)} of .
The nomenclature comes from the fact that when , we have , the ordinary Stirling number of the second kind.
Explicit expressions for, and identities satisfied by, the have been
obtained by numerous authors, and combinatorial interpretations have been
presented. Here we provide a new combinatorial interpretation that retains the
spirit of the familiar interpretation of as a count of
partitions. Specifically, we associate to each a quasi-threshold graph
, and we show that enumerates partitions of the vertex set of
into classes that do not span an edge of . We also discuss some
relatives of, and consequences of, our interpretation, including -analogs
and bijections between families of labelled forests and sets of restricted
partitions.Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.00
Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube
We study a two-parameter generalization of the Catalan numbers:
is the number of ways to subdivide the -dimensional hypercube into
rectangular blocks using orthogonal partitions of fixed arity . Bremner \&
Dotsenko introduced in their work on Boardman--Vogt tensor
products of operads; they used homological algebra to prove a recursive formula
and a functional equation. We express as simple finite sums, and
determine their growth rate and asymptotic behaviour. We give an elementary
proof of the functional equation, using a bijection between hypercube
decompositions and a family of full -ary trees. Our results generalize the
well-known correspondence between Catalan numbers and full binary trees
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