We prove that any IN-rational sequence s = (sn) n1 of non-negative integers satisfying the Kraft strict inequality P n1 snk ?n < 1 is the enumerative sequence of leaves by height of a rational k-ary tree. Particular cases of this result had been previously proven. We give some partial results in the equality case.

In this paper we compute the number of reduced decompositions of certain permutations ∈ Sn as a product of transpositions (1; 2), (1; 3), : : :, (1; n). We present several combinatorial correspondences between these decompositions and combinatorial objects such as Catalan paths and k-ary trees. c © 1999 Elsevier Science B.V. All rights reserved

fuzzy rough n-ary subhypergroups are introduced and characterized.

In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees.

We devise a condition strictly between the existence of an $n$-ary and $n{+}1$-ary near-unanimity term. evaluate exactly distributivity modularity levels implied by such condition.

In this paper, by using the notion of fuzzy subsets, the concept of F-permutation is introduced. Then by applying this notion the concepts of presentation of an F-polygroup, graph of an F-permutation and F-permutation matrices are investigated.