Geodesic
Record statistics in integer compositions
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009).

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A $\textit{composition}$ $\sigma =a_1 a_2 \ldots a_m$ of $n$ is an ordered collection of positive integers whose sum is $n$. An element $a_i$ in $\sigma$ is a strong (weak) $\textit{record}$ if $a_i> a_j (a_i \geq a_j)$ for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all compositions of $n$, as well as for the sum of the positions of the records in all compositions of $n$, where the parts $a_i$ belong to a fixed subset $A$ of the natural numbers. In particular when $A=\mathbb{N}$, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of $n$. 
@article{DMTCS_2009_special_256_a13,
     author = {Knopfmacher, Arnold and Mansour, Toufik},
     title = {Record statistics in integer compositions},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)},
     year = {2009},
     doi = {10.46298/dmtcs.2691},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2691/}
}
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Knopfmacher, Arnold; Mansour, Toufik. Record statistics in integer compositions. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009) (2009). doi : 10.46298/dmtcs.2691. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2691/

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