The enumeration of integer sequences with a given number of colored records (Q1209655)

An integer-valued discrete time series \(W\) with domain \(\{1,\dots,n\}\) has a record at time \(i\) if \(W(i)>W(j)\) for all \(j<i\). Counting time series with a given number of records can be achieved using Stirling numbers. The author introduces a colouring scheme for the state space, replacing the role of Stirling numbers by general factorial numbers (see for example the author's paper in [Congr. Numerantium 77, 187-194 (1990)] for a discussion of such numbers). The colouring highlights `head and tail records', where the tail of a time series begins after the first occurrence of the maximum. The tail records are of interest in testing ranks for randomness; some discussion involving simulation results ends the paper.