Counting descents, rises, and levels, with prescribed first element, in words

Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper were extended and generalized in several ways. In this paper, we shall fix a set partition of the natural numbers $N$, $(N_1, ..., N_t)$, and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in $N_i$ over the set of words over the alphabet $[k]$. In particular, we refine and generalize some of the results in [Counting occurrences of some subword patterns, Discrete Mathematics and Theoretical Computer Science 6 (2003), 001-012.].Comment: 20 pages, sections 3 and 4 are adde

Counting descents, rises, and levels, with prescribed first element, in words

Combinatoric

Lauki sareko patroi bitarren kalkulua, oinarrizko konbinatoriaren eskutik

This work is a seque! of paper Yurramendi (20 13), where so me mathematical formulae for counting the number of bi nary rectangu lar grids have been obtained. The aim of this work is the combinatoria] deduction of those mathematical formulae. Two matrices are considered equivalen! if they are the same after reflection, 180 degree rotation ora combination of both. We experi mentally conjecture the formu lae with the help of a computational algorithm. The problem is classified according to the parity of the number of rows and colum11s of the matrices. Further results are added.; Lan hau Yu1nmendi (20 13) artikuluaren ondorioa da eta bertan formula matematiko batzuk lortu dira lauki angeluzuzen bitar kopu rua kalku latzeko. Lan honen helburua formula matematiko horiek konbinatoriaren bidez deduzitzea da. Bi matrize baliokide dira baldin eta berdinak badira is lapen bat, 180 graduko biraketa edo bien konbinaketaren ondoren. Konputaz io a lgoritmo baten laguntzaz zenbait fo rmularen aieruak egin ditugu esperimentalki Problema ebatzi da matrizeen !erro eta zutabe kopuruen arteko paritatearen arabera. Gero, zenbait emaitza gehitu dira