The largest missing value in a composition of an integer

Archibald and Knopfmacher recently considered the largest missing value in a composition of an integer and established mean and variance. Our alternative, probabilistic approach produces (in principle) all moments. In order to show that our forms match the ones given by Archibald and Knopfmacher, we have to derive some identities which are interesting in itself. It is a one parameter family of identities, and the first one is (equivalent to) the celebrated identity by Allouche and Shallit. We finally provide a simple direct analysis of the LMV case: if the largest missing value is exactly one smaller than the largest value, we say that the sequence has the LMV property.

• Type: Journal
• Year: 2011
• Where: DM
• Authors: Margaret Archibald, Arnold Knopfmacher
• Keywords: DM 2011, Education, Identities, Probabilistic Approach, Simple Direct Analysis