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A Classic Proof of a Recurrence for a Very Classical Sequence (1997)
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| Citations: | 18 - 0 self |
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@MISC{Foata97aclassic,
author = {Dominique Foata and Doron Zeilberger},
title = {A Classic Proof of a Recurrence for a Very Classical Sequence},
year = {1997}
}
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Abstract
Richard Stanley [St96] has recently narrated the fascinating story of how the classical Schroder [Sch1870] numbers s(n) are even more classical than was previously believed. They (at least s(10)=103049) were known to Hipparchus (190 127 b.c.). Stanley recalled the three-term linear recurrence ([Co64; Co74, p. 57]) 3(2n&1) s(n)=(n+1) s(n+1)+(n&2) s(n&1) (n 2), s(1)=s(2)=1, (1) and stated that ``no direct combinatorial proof of this formula seems to be known.' ' The purpose of this note is to fill this gap. The present proof reflects the ideas of our great master, M.-P. Schutzenberger (1920 1996), who taught us that every algebraic relation is to be given a combinatorial counterpart and vice versa. This methodology has been vigorously and successfully pursued by the Ecole bordelaise (e.g., [Cor75, Vi85].) * E-mail: foata math.u-strasbg.fr.
