2,771 research outputs found
Counting and Matching
Lists, multisets and partitions are fundamental datatypes in mathematics and computing. There are basic transformations from lists to multisets (called "accumulation") and also from lists to partitions (called "matching"). We show how these transformations arise systematically by forgetting/abstracting away certain aspects of information, namely order (transposition) and identity (substitution). Our main result is that suitable restrictions of these transformations are isomorphisms: This reveals fundamental correspondences between elementary datatypes. These restrictions involve "incremental" lists/multisets and "non-crossing" partitions/lists. While the process of forgetting information can be precisely spelled out in the language of category theory, the relevant constructions are very combinatorial in nature. The lists, partitions and multisets in these constructions are counted by Bell numbers and Catalan numbers. One side-product of our main result is a (terminating) rewriting system that turns an arbitrary partition into a non-crossing partition, without improper nestings
On Multidimensional Inequality in Partitions of Multisets
We study multidimensional inequality in partitions of finite multisets with thresholds. In such a setting, a Lorenz-like preorder, a family of functions preserving such a preorder, and a counterpart of the Pigou-Dalton transfers are defined, and a version of the celebrated Hardy-Littlewood-Pölya characterization results is provided.Multisets, majorization, Lorenz preorder, Hardy-Littlewood-Polya theorem, transfers
Reconstructing multisets over commutative groupoids and affine functions over nonassociative semirings
A reconstruction problem is formulated for multisets over commutative
groupoids. The cards of a multiset are obtained by replacing a pair of its
elements by their sum. Necessary and sufficient conditions for the
reconstructibility of multisets are determined. These results find an
application in a different kind of reconstruction problem for functions of
several arguments and identification minors: classes of linear or affine
functions over nonassociative semirings are shown to be weakly reconstructible.
Moreover, affine functions of sufficiently large arity over finite fields are
reconstructible.Comment: 18 pages. Int. J. Algebra Comput. (2014
Symmetric Group Character Degrees and Hook Numbers
In this article we prove the following result: that for any two natural
numbers k and j, and for all sufficiently large symmetric groups Sym(n), there
are k disjoint sets of j irreducible characters of Sym(n), such that each set
consists of characters with the same degree, and distinct sets have different
degrees. In particular, this resolves a conjecture most recently made by
Moret\'o. The methods employed here are based upon the duality between
irreducible characters of the symmetric groups and the partitions to which they
correspond. Consequently, the paper is combinatorial in nature.Comment: 24 pages, to appear in Proc. London Math. So
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