Geodesic
A Sequential Search Distribution: Proofreading, Russian Roulette, and the Incomplete q-Eulerian Polynomials
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001), DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001) (2001).

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The distribution for the number of searches needed to find k of n lost objects is expressed in terms of a refinement of the q-Eulerian polynomials, for which formulae are developed involving homogeneous symmetric polynomials. In the case when k=n and the find probability remains constant, relatively simple and efficient formulas are obtained.From our main theorem, we further (1) deduce the inverse absorption distribution and (2) determine the expected number of times the survivor pulls the trigger in an n-player game of Russian roulette. 
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     author = {Herbranson, Travis and Rawlings, Don},
     title = {A {Sequential} {Search} {Distribution:} {Proofreading,} {Russian} {Roulette,} and the {Incomplete} {q-Eulerian} {Polynomials}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
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     year = {2001},
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Herbranson, Travis; Rawlings, Don. A Sequential Search Distribution: Proofreading, Russian Roulette, and the Incomplete q-Eulerian Polynomials. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001), DMTCS Proceedings vol. AA, Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG 2001) (2001). doi : 10.46298/dmtcs.2281. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2281/

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