## Abstract

Let s, t be any numbers in {0, 1} and let π = π1π2 · · · πm be any word,

we say that i ∈ [m − 1] is an (s, t) parity-rise if πi ≡ s (mod 2), πi+1 ≡ t (mod 2)

whenever πi < πi+1. We denote the number occurrences of (s, t) parity-rises in π by risest(π). Also, we denote the total sizes of the (s, t) parity-rises in π by sizest(π), that is, sizest(π) = ∑πi<πi+1 (πi+1 − πi). A composition π = π1π2 · · · πm of a positive integer n is an ordered collection of one or more positive integers whose sum is n. The number of summands, namely m, is called the number of parts of π. In this paper, by using tools of linear algebra, we found the generating function that count the number of all compositions of n with m parts according to the statistics risest and sizest, for all s, t.

we say that i ∈ [m − 1] is an (s, t) parity-rise if πi ≡ s (mod 2), πi+1 ≡ t (mod 2)

whenever πi < πi+1. We denote the number occurrences of (s, t) parity-rises in π by risest(π). Also, we denote the total sizes of the (s, t) parity-rises in π by sizest(π), that is, sizest(π) = ∑πi<πi+1 (πi+1 − πi). A composition π = π1π2 · · · πm of a positive integer n is an ordered collection of one or more positive integers whose sum is n. The number of summands, namely m, is called the number of parts of π. In this paper, by using tools of linear algebra, we found the generating function that count the number of all compositions of n with m parts according to the statistics risest and sizest, for all s, t.

Original language | English |
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Article number | Article 5 |

Number of pages | 15 |

Journal | Online Journal of Analytic Combinatorics |

Volume | 10 |

State | Published - 2015 |