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The Bruhat rank of a binary symmetric staircase pattern

Zhibin Du, Carlos M. da Fonseca (2016)

Open Mathematics

In this work we show that the Bruhat rank of a symmetric (0,1)-matrix of order n with a staircase pattern, total support, and containing In, is at most 2. Several other related questions are also discussed. Some illustrative examples are presented.

The Milgram non-operad

Michael Brinkmeier (1999)

Annales de l'institut Fourier

C. Berger claimed to have constructed an E n -operad-structure on the permutohedras, whose associated monad is exactly the Milgram model for the free loop spaces. In this paper I will show that this statement is not correct.

The ordering of commutative terms

Jaroslav Ježek (2006)

Czechoslovak Mathematical Journal

By a commutative term we mean an element of the free commutative groupoid F of infinite rank. For two commutative terms a , b write a b if b contains a subterm that is a substitution instance of a . With respect to this relation, F is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other things, we prove that every commutative term (or its block in the factor) is a definable element. Consequently, the ordered set has...

Transitivity and partial order

Jiří Klaška (1997)

Mathematica Bohemica

In this paper we find a one-to-one correspondence between transitive relations and partial orders. On the basis of this correspondence we deduce the recurrence formula for enumeration of their numbers. We also determine the number of all transitive relations on an arbitrary n -element set up to n = 14 .

Two results on a partial ordering of finite sequences

Martin Klazar (1993)

Commentationes Mathematicae Universitatis Carolinae

In the first part of the paper we are concerned about finite sequences (over arbitrary symbols) u for which E x ( u , n ) = O ( n ) . The function E x ( u , n ) measures the maximum length of finite sequences over n symbols which contain no subsequence of the type u . It follows from the result of Hart and Sharir that the containment a b a b a u is a (minimal) obstacle to E x ( u , n ) = O ( n ) . We show by means of a construction due to Sharir and Wiernik that there is another obstacle to the linear growth. In the second part of the paper we investigate whether...

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