A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics

From the output produced by a memoryless deletion channel from a uniformly random input of known length $n$, one obtains a posterior distribution on the channel input. The difference between the Shannon entropy of this distribution and that of the uniform prior measures the amount of information about the channel input which is conveyed by the output of length $m$, and it is natural to ask for which outputs this is extremized. This question was posed in a previous work, where it was conjectured on the basis of experimental data that the entropy of the posterior is minimized and maximized by the constant strings $\texttt{000}\ldots$ and $\texttt{111}\ldots$ and the alternating strings $\texttt{0101}\ldots$ and $\texttt{1010}\ldots$ respectively. In the present work we confirm the minimization conjecture in the asymptotic limit using results from hidden word statistics. We show how the analytic-combinatorial methods of Flajolet, Szpankowski and Vall\'ee for dealing with the hidden pattern matching problem can be applied to resolve the case of fixed output length and $n\rightarrow\infty$, by obtaining estimates for the entropy in terms of the moments of the posterior distribution and establishing its minimization via a measure of autocorrelation.Comment: 11 pages, 2 figure

Combinatorics of patience sorting piles

Despite having been introduced in 1962 by C.L. Mallows, the combinatorial algorithm Patience Sorting is only now beginning to receive significant attention due to such recent deep results as the Baik-Deift-Johansson Theorem that connect it to fields including Probabilistic Combinatorics and Random Matrix Theory. The aim of this work is to develop some of the more basic combinatorics of the Patience Sorting Algorithm. In particular, we exploit the similarities between Patience Sorting and the Schensted Insertion Algorithm in order to do things that include defining an analog of the Knuth relations and extending Patience Sorting to a bijection between permutations and certain pairs of set partitions. As an application of these constructions we characterize and enumerate the set S_n(3-\bar{1}-42) of permutations that avoid the generalized permutation pattern 2-31 unless it is part of the generalized pattern 3-1-42.Comment: 19 pages, LaTeX; uses pstricks; view PS, not DVI; use dvips + ps2pdf, not dvi2pdf; part of FPSAC'05 proceedings; v3: final journal version, revised Section 3.