Canonical Representatives Research Articles

Remark on the simplicial-cosimplicial tensor product

We show that the existence of canonical representatives for the elements of the tensor product (coend) of a simplicial and a cosimplicial set depends only on the Eilenberg-Zilber property of the given cosimplicial set. Thus the second condition which is used in [5] for achieving this result is superfluous.

Computation of threshold function Chow parameters by graphical means †

Abstract Existing methods of computing Chow parameter tabulations for threshold functions generally employ a complex generation algorithm based on a 2-monotonic lattice. It is here shown that the Torng map is a preferable form of this lattice, and that a simple generation algorithm based on self-duality can be adopted to generate the positive canonical threshold function representatives. Translation of these representative functions into appropriate minimum-integer weights and thresholds then produces the desired standard Chow parameter tabulations. The complete procedure has little complexity and is ideally suitable for hand computation: its latter limitation is when the volume of the data becomes too great to hand-compile.

Majority decision functions of up to six variables

This paper defines the canonical representative of each equivalence class in the classification of the majority decision functions by complementing and permuting variables and by complementing the output. Also, a method is proposed to obtain all the representatives with their optimum structures, and a table of the representatives of the majority decision functions of up to six variables is provided. The reader should be familiar with the content of a previous paper by the authors, included as reference [1].

Two examples of unbalanced Wilf-equivalence

We prove that the set of patterns {1324,3416725} is Wilf-equivalent to the pattern 1234 and that the set of patterns {2143,3142,246135} is Wilf-equivalent to the set of patterns {2413,3142}. These are the first known unbalanced Wilf-equivalences for classical patterns between finite sets of patterns. A pattern is an equivalence class of sequences under order-isomorphism. Two sequences �1 and �2 over totally ordered alphabets are order-isomorphic if, for any pair of positions i and j, we have �1(i) < �1(j) if and only if �2(i) < �2(j). We identify a pattern with its canonical representative, in which the kth smallest letter is k. We say that a permutation � contains a patternifhas a subsequence order-isomorphic to �, otherwise we say that � avoids �. We denote the set of permutations of length n by Sn, the set of permutations in Sn avoiding patternby Avn(�), and the set of permutations in Sn avoiding every pattern in a setby Avn(�). We say that two (sets of) patterns � ' and � '' are Wilf-equivalent, denoted � ' ∼ � '' , if |Avn(� ' )| = |Avn(� '' )| for all n ∈ N. We call a Wilf-equivalence unbalanced if the two sets of patterns do not contain the same number of patterns of each length. We will sometimes talk about the type of an unbalanced Wilf-equivalence, defined by the lengths of the patterns: for example, the two Wilf-equivalences proved in this paper have types (4) ∼ (4,7) and (4,4) ∼ (4,4,6). In Section 1, we prove that {1324,3416725} ∼ 1234. In Section 2, we prove that {2143,3142,246135} ∼ {2413,3142}. In Section 3, we conjecture a few other unbalanced Wilf-equivalences.