10.46298/dmtcs.2156 Shabani, Armend Armend Shabani Gjergji, Rexhep Rexhep Gjergji Statistics for 3-letter patterns with repetitions in compositions episciences.org 2016 Cramer’s method Subwords generating functions [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] contact@episciences.org episciences.org 2015-05-05T00:00:00+02:00 2021-08-23T23:09:16+02:00 2016-05-31 en Journal article https://dmtcs.episciences.org/2156 https://hal.archives-ouvertes.fr/hal-01352850v1 1365-8050 PDF 1 Discrete Mathematics & Theoretical Computer Science ; Vol. 17 no. 3 ; Combinatorics ; 1365-8050 International audience A composition $\pi = \pi_1 \pi_2 \cdots \pi_m$ 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 $\pi$. Using linear algebra, we determine formulas for generating functions that count compositions of $n$ with $m$ parts, according to the number of occurrences of the subword pattern $\tau$, and according to the sum, over all occurrences of $\tau$, of the first integers in their respective occurrences, where $\tau$ is any pattern of length three with exactly 2 distinct letters.