Partially Ordered Generalized Patterns and <Emphasis Type="Italic">k</Emphasis>-ary Words

Recently, Kitaev [9] introduced partially ordered generalized patterns (POGPs) in the symmetric group, which further generalize the generalized permutation patterns introduced by Babson and Steingrímsson [1]. A POGP p is a GP some of whose letters are incomparable. In this paper, we study the generating functions (g.f.) for the number of k-ary words avoiding some POGPs. We give analogues, extend and generalize several known results, as well as get some new results. In particular, we give the g.f. for the entire distribution of the maximum number of non-overlapping occurrences of a pattern p with no dashes (which is allowed to have repetition of letters), provided we know the g.f. for the number of k-ary words that avoid p.AMS Subject Classification: 05A05, 05A15.     

Coloured Permutations Containing and Avoiding Certain Patterns

Let be the set of all coloured permutations on the symbols 1, 2, . . . , n with colours 1, 2, . . . , r, which is the analogous of the symmetric group when r = 1, and the hyperoctahedral group when r = 2. Let be a subset of d colours; we define to be the set of all coloured permutations . We prove that the number of -avoiding coloured permutations in . We then prove that for any , the number of coloured permutations in which avoid all patterns in except for and contain exactly once equals . Finally, for any , this number equals . These results generalize recent results due to Mansour, Mansour and West, and Simion.AMS Subject Classification: 05A05, 05A15.     

The Problem of the Pawns

In this paper we study the number M _m;n of ways to place nonattacking pawns on an m x n chessboard. We find an upper bound for M _m;n and consider a lower bound for M _m;n by reducing this problem to that of tiling an (m+1)x(n+1) board with square tiles of size 1x1 and 2x2. Also, we use the transfer-matrix method to implement an algorithm that allows us to get an explicit formula for M _m;n for given m. Moreover, we show that the double limit := lim _m;n (M _m;n ) ^1/mn exists and 2.25915263 n 2.26055675. Also, the true value of n is around 2.2591535382327...AMS Subject Classification: 05A16, 05C50, 52C20, 82B20.     

Noncrossing Normal Ordering for Functions of Boson Operators

Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel (Lett. Nuovo Cimento 10(13):565–567, 1974), in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (a ^r (a ^† ) ^s ) ⁿ ) and (a ^r +(a ^† ) ^s ) ⁿ , plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.     

A polynomial generalization of some associated sequences related to set partitions

Periodica Mathematica Hungarica - In this paper, we consider a common polynomial generalization, denoted by $$w_m(n,k)=w_m^{a,b,c,d}(n,k)$$ , of several types of associated sequences. When $$a=0$$...     

避免$2$字母符号模式的具有偶数个符号的带符号排列

Let Dn be the set of all signed permutations on [n] = {1,... ,n} with even signs, and let ：Dn（T） be the set of all signed permutations in Dn which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets Dn（T） where T B2. Some of the cardinalities encountered involve inverse binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers.     

Exposing Deep Representations to a Recurrent Expansion with Multiple Repeats for Fuel Cells Time Series Prognosis

The green conversion of proton exchange membrane fuel cells (PEMFCs) has received particular attention in both stationary and transportation applications. However, the poor durability of PEMFC represents a major problem that hampers its commercial application since dynamic operating conditions, including physical deterioration, have a serious impact on the cell performance. Under these circumstances, prognosis and health management (PHM) plays an important role in prolonging durability and preventing damage propagation via the accurate planning of a condition-based maintenance (CBM) schedule. In this specific topic, health deterioration modeling with deep learning (DL) is the widely studied representation learning tool due to its adaptation ability to rapid changes in data complexity and drift. In this context, the present paper proposes an investigation of further deeper representations by exposing DL models themselves to recurrent expansion with multiple repeats. Such a recurrent expansion of DL (REDL) allows new, more meaningful representations to be explored by repeatedly using generated feature maps and responses to create new robust models. The proposed REDL, which is designed to be an adaptive learning algorithm, is tested on a PEMFC deterioration dataset and compared to its deep learning baseline version under time series analysis. Using multiple numeric and visual metrics, the results support the REDL learning scheme by showing promising performances.