On the distribution of some Euler-Mahonian statistics

We give a direct combinatorial proof of the equidistribution of two pairs of permutation statistics, (des, aid) and (lec, inv), which have been previously shown to have the same joint distribution as (exc, maj), the major index and the number of excedances of a permutation. Moreover, the triple (pix, lec, inv) was shown to have the same distribution as (fix, exc, maj), where fix is the number of fixed points of a permutation. We define a new statistic aix so that our bijection maps (pix, lec, inv) to (aix, des, aid). We also find an Eulerian partner das for a Mahonian statistic mix defined using mesh patterns, so that (das, mix) is equidistributed with (des, inv).Comment: 9 page

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.

A decision support tool for supporting individuals living with long-term conditions make informed choices: LTC-Choices tool for continuous healthcare

An increasing number of individuals are now living with some form of chronic, long-term condition (LTC). The holistic perspective of LTCs makes it important to acknowledge that priorities and decisions are in fluctuation over the course of an individual’s life. The landscape of digital healthcare is full of information systems that capture individuals’ health data, clinical guidelines and/or advice on health conditions, which taken together can help create a comprehensive overview of suitable lifestyle choices to optimise health and well-being. Despite this, there is no evidence of existing frameworks to support individuals living with LTCs from a continuum of care perspective. In this paper, we propose such a multidimensional model for a decision support tool – LTC-Choices. This tool was developed from existing work conducted by the authors around use of multicriteria to support health decisionmaking. We illustrate how LTC-Choices can be implemented using the example of individuals living post-stroke