Geodesic
Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12) (2012).

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We define the notion of $t$-free for locally restricted compositions, which means roughly that if such a composition contains a part $c_i$ and nearby parts are at least $t$ smaller, then $c_i$ can be replaced by any larger part. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part and number distinct parts, all accurate to $o(1)$. 
@article{DMTCS_2012_special_262_a18,
     author = {Bender, Edward and Canfield, Rodney and Gao, Zhicheng},
     title = {Locally {Restricted} {Compositions} {IV.} {Nearly} {Free} {Large} {Parts} and {Gap-Freeness}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)},
     year = {2012},
     doi = {10.46298/dmtcs.2997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2997/}
}
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Bender, Edward; Canfield, Rodney; Gao, Zhicheng. Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), DMTCS Proceedings vol. AQ, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12) (2012). doi : 10.46298/dmtcs.2997. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2997/

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