A maxdrop statistic for standard Young tableaux

In this paper, we introduce a new statistic on standard Young tableaux that is closely related to the maxdrop permutation statistic that was introduced by the first author. We prove that the value of the statistic must be attained at one of the corners of the standard Young tableau. We determine the coefficients of the generating function of this statistic over two-row standard Young tableaux having [Formula: see text] cells. We prove several results for this new statistic that include unimodality of the coefficients for the two-row case.

Spatiotemporal patterns and agroecological risk factors for cutaneous and renal glomerular vasculopathy (Alabama Rot) in dogs in the UK

The annual outbreaks of cutaneous and renal glomerular vasculopathy (CRGV) reported in UK dogs display a distinct seasonal pattern (November to May) suggesting possible climatic drivers of the disease. The objectives of this study were to explore disease clustering and identify associations between agroecological factors and CRGV occurrence. Kernel-smoothed maps were generated to show the annual reporting distribution of CRGV, Kuldorff’s space–time permutation statistic used to identify significant spatiotemporal case clusters and a boosted regression tree model developed to quantify associations between CRGV case locations and a range of agroecological factors. The majority of diagnoses (92 per cent) were reported between November and May while the number of regions reporting the disease increased between 2012 and 2017. Two significant spatiotemporal clusters were identified—one in the New Forest during February and March 2013, and one adjacent to it (April 2015 to May 2017)—showing significantly higher and lower proportions of cases than the rest of the UK, respectively, for the indicated time periods. A moderately significant high-risk cluster (P=0.087) was also identified in the Manchester area of northern England between February and April 2014. Habitat was the predictor with the highest relative contribution to CRGV distribution (20.3 per cent). Cases were generally associated with woodlands, increasing mean maximum temperatures in winter, spring and autumn, increasing mean rainfall in winter and spring and decreasing cattle and sheep density. Understanding of such factors may help develop causal models for CRGV occurrence.

On the Parity of Certain Coefficients for a $q$-Analogue of the Catalan Numbers

The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, $C_n$, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan. In particular, it is well known that $C_n$ is odd if and only if $n = 2^k-1$ for some $k \geq 0$. The polynomial $F_n^{ch}(321;q) = \sum_{\sigma \in Av_n(321)} q^{ch(\sigma)}$, where $Av_n(321)$ is the set of permutations in $S_n$ that avoid 321 and $ch$ is the charge statistic, is a $q$-analogue of the Catalan numbers since specializing $q=1$ gives $C_n$. We prove that the coefficient of $q^i$ in $F_{2^k-1}^{ch}(321;q)$ is even if $i \geq 1$, giving a refinement of the "if" direction of the $C_n$ parity result. Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index. In addition, Sagan and Savage have recently defined a notion of $st$-Wilf equivalence for any permutation statistic $st$ and any two sets of permutations $\Pi$ and $\Pi'$. We say $\Pi$ and $\Pi'$ are $st$-Wilf equivalent if $\sum_{\sigma \in Av_n(\Pi)} q^{st(\sigma)} = \sum_{\sigma \in Av_n(\Pi')} q^{st(\sigma)}$. In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of $S_3$.

Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

An Interesting New Mahonian Permutation Statistic

The standard algorithm for generating a random permutation gives rise to an obvious permutation statistic DIS that is readily seen to be Mahonian. We give evidence showing that it is not equal to any previously published statistic. Nor does its joint distribution with the standard Eulerian statistics des and exc appear to coincide with any known Euler-Mahonian pair. A general construction of Skandera yields an Eulerian partner eul such that (eul, DIS) is equidistributed with (des, MAJ). However eul itself appears not to be a known Eulerian statistic. Several ideas for further research on this topic are listed.

Counting descents, rises, and levels, with prescribed first element, in words

Combinatorics International audience Recently, Kitaev and Remmel refined the well-known permutation statistic "descent" by fixing parity of one of the descent's numbers which was extended and generalized in several ways in the literature. In this paper, we shall fix a set partition of the natural numbers N,(N1, ..., Ns), and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = 1, ..., k. In particular, we refine and generalize some of the results by Burstein and Mansour

Classifying Descents According to Equivalence mod k

In an earlier paper the authors refine the well-known permutation statistic "descent" by fixing parity of (exactly) one of the descent's numbers. In the current paper, we generalize the results of that earlier paper by studying descents according to whether the first or the second element in a descent pair is divisible by $k$ for some $k\geq 2$. We provide either an explicit or an inclusion-exclusion type formula for the distribution of the new statistics. Based on our results we obtain combinatorial proofs of a number of remarkable identities. We also provide bijective proofs of some of our results and state a number of open problems.

Calcul Basique des Permutations Signées, II: Analogues Finis des Fonctions de BesseL

The traditional basic calculus on permutation statistic distributions is extended to the case of signed permutations. This provides with a combinatorial interpretation of the basic Bessel functions and their finite analogues. Le calcul basique classique sur les distributions des statistiques des permutations est prolongé au cas des permutations signées. Ce calcul permet ainsi de donner une interprétation combinatoire aux fonctions basiques de Bessel et à leurs analogues finis.