The Largest Missing Value in a Composition of an Integer and Some Allouche-Shallit-Type Identities
Université Libre de Bruxelles
Département d’Informatique, CP 212
Boulevard du Triomphe
University of Stellenbosch
Archibald and Knopfmacher recently considered the largest missing value
in a composition of an integer and established the mean and variance.
Our alternative, probabilistic approach produces (in principle)
moments in an almost automatic way. In order to show that
our forms match the ones given by Archibald and Knopfmacher, we have to
derive some identities which are interesting on their own. We construct
a one-parameter family of identities, and the first one is (equivalent to)
the celebrated identity due to Allouche and Shallit. We finally provide
a simple direct analysis of the LMV(-1) case: if the largest
missing value is exactly one smaller than the largest value, we say
that the sequence has the LMV(-1) property.
Full version: pdf,
(Concerned with sequence
Received April 18 2012;
revised version received October 25 2012.
Published in Journal of Integer Sequences, March 3 2013.
Journal of Integer Sequences home page