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《Journal of Pure and Applied Algebra》2023,227(2):107190
Let W be a finite Coxeter group and X a subset of W. The length polynomial is defined by , where ? is the length function on W. If then we call the involution length polynomial of W. In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, and the involution length polynomial, in any finite Coxeter group W. In particular, these results correct errors in [11] for the involution length polynomials of Coxeter groups of type and . Moreover, we give a counterexample to a unimodality conjecture stated in [11]. 相似文献
3.
Let Π = B_1/B_2/… /B_k be any set partition of[n]= {1,2,...,n} satisfying that entries are increasing in each block and blocks are arranged in increasing order of their first entries.Then Callan defined the flattened Π to be the permutation of[n]obtained by erasing the divers between its blocks,and Callan also enumerated the number of set partitions of[n]whose flattening avoids a single3-letter pattern.Mansour posed the question of counting set partitions of[n]whose flattening avoids a pattern of length 4.In this paper,we present the number of set partitions of[n]whose flattening avoids one of the patterns:1234,1243,1324,1342,1423,1432,3142 and 4132. 相似文献
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Eva Y.P. Deng 《Discrete Mathematics》2009,309(12):4108-4115
Let Ak be the set of permutations in the symmetric group Sk with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns Ak. We present a bijection between symmetric Schröder paths of length 2n and involutions of length n+1 avoiding A4. Statistics such as the number of right-to-left maxima and fixed points of the involution correspond to the number of steps in the symmetric Schröder path of a particular type. For each k≥3 we determine the generating function for the number of involutions avoiding the subsequences in Ak, according to length, first entry and number of fixed points. 相似文献
6.
We show that the principal order ideal of an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this
property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin
paths is demonstrated.
This article is largely based on results from the second author’s M.Sc. thesis [15]. 相似文献
7.
In this short note, we focus on self-inverse Sheffer sequences and involutions in the Riordan group. We translate the results of Brown and Kuczma on self-inverse sequences of Sheffer polynomials to describe all involutions in the Riordan group. 相似文献
8.
Expansions in terms of Bessel functions are considered of the Kummer function 1
F
1(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials
in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables
are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed
together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the
expansions is explained for large values of the parameter a of the Kummer function. 相似文献
9.
Jan Adem Yves Crama Willy Gochet Frits C.R. Spieksma 《Discrete Applied Mathematics》2008,156(13):2459-2468
We propose a generic model for the “weighted voting” aggregation step performed by several methods in supervised classification. Further, we construct an algorithm to enumerate the number of distinct aggregate classifiers that arise in this model. When there are only two classes in the classification problem, we show that a class of functions that arises from aggregate classifiers coincides with the class of self-dual positive threshold Boolean functions. 相似文献
10.
Toufik Mansour 《Journal of Difference Equations and Applications》2013,19(1):16-36
We prove that the generating function for the number of flattened permutations having a given number of occurrences of the pattern 13-2 is rational, by using the recurrence relations and the kernel method. 相似文献
11.
R.M. Green 《Journal of Algebraic Combinatorics》2002,15(3):241-252
We introduce the notion of 321-avoiding permutations in the affine Weyl group W of type A
n – 1 by considering the group as a George group (in the sense of Eriksson and Eriksson). This enables us to generalize a result of Billey, Jockusch and Stanley to show that the 321-avoiding permutations in W coincide with the set of fully commutative elements; in other words, any two reduced expressions for a 321-avoiding element of W (considered as a Coxeter group) may be obtained from each other by repeated applications of short braid relations.Using Shi's characterization of the Kazhdan–Lusztig cells in the group W, we use our main result to show that the fully commutative elements of W form a union of Kazhdan–Lusztig cells. This phenomenon has been studied by the author and J. Losonczy for finite Coxeter groups, and is interesting partly because it allows certain structure constants for the Kazhdan–Lusztig basis of the associated Hecke algebra to be computed combinatorially.We also show how some of our results can be generalized to a larger group of permutations, the extended affine Weyl group associated to GL
n
() 相似文献
12.
《Indagationes Mathematicae》2022,33(6):1221-1235
In a recent paper (Temme, 2021) new asymptotic expansions are given for the Kummer functions and for large positive values of and , with fixed and special attention for the case . In this paper we extend the approach and also accept large values of . The new expansions are valid when at least one of the parameters , , or is large. We provide numerical tables to show the performance of the expansions. 相似文献
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《Discrete Mathematics》2020,343(11):112043
The notion of a Riordan graph was introduced recently, and it is a far-reaching generalization of the well-known Pascal graphs and Toeplitz graphs. However, apart from a certain subclass of Toeplitz graphs, nothing was known on independent sets in Riordan graphs.In this paper, we give exact enumeration and lower and upper bounds for the number of independent sets for various classes of Riordan graphs. Remarkably, we offer a variety of methods to solve the problems that range from the structural decomposition theorem to methods in combinatorics on words. Some of our results are valid for any graph. 相似文献
14.
Bracketed words are basic structures both in mathematics (such as Rota-Baxter algebras) and mathematical physics (such as rooted trees) where the locations of the substructures are important. In this paper, we give the classification of the relative locations of two bracketed subwords of a bracketed word in an operated semigroup into the separated, nested, and intersecting cases. We achieve this by establishing a correspondence between relative locations of bracketed words and those of words by applying the concept of Motzkin words which are the algebraic forms of Motzkin paths. 相似文献
15.
Gerhard Dorfer Wilfried Meidl Arne Winterhof 《Finite Fields and Their Applications》2004,10(4):113-652
Recently, Dorfer and Winterhof introduced and analyzed a lattice test for sequences of length n over a finite field. We determine the number of sequences η of length n with given largest dimension Sn(η)=S for passing this test. From this result we derive an exact formula for the expected value of Sn(η). For the binary case we characterize the (infinite) sequences η with maximal possible Sn(η) for all n. 相似文献
16.
Feng Qi Run-Qing Cui Bai-Ni Guo 《Journal of Mathematical Analysis and Applications》2005,310(1):303-308
By using the first Binet's formula the strictly completely monotonic properties of functions involving the psi and polygamma functions are obtained. As direct consequences, two inequalities are proved. As an application, the best lower and upper bounds of the nth harmonic number are established. 相似文献
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Joel Brewster Lewis 《Journal of Combinatorial Theory, Series A》2011,118(4):1436-1450
We define a class Ln,k of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give bijections between the set A2n(1234) of alternating permutations of length 2n with no four-term increasing subsequence and standard Young tableaux of shape 〈n3〉, and between the set A2n+1(1234) and standard Young tableaux of shape 〈3n−1,2,1〉. This represents the first enumeration of alternating permutations avoiding a pattern of length four. We also extend previous work on doubly-alternating permutations (alternating permutations whose inverses are alternating) to our more general context.The set Ln,k may be viewed as the set of reading words of the standard Young tableaux of a certain skew shape. In the last section of the paper, we expand our study to consider pattern avoidance in the reading words of standard Young tableaux of any skew shape. We show bijectively that the number of standard Young tableaux of shape λ/μ whose reading words avoid 213 is a natural μ-analogue of the Catalan numbers (and in particular does not depend on λ, up to a simple technical condition), and that there are similar results for the patterns 132, 231 and 312. 相似文献
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Radoslav Fulek 《Discrete Mathematics》2009,309(6):1736-2410
In this note by saying that a 0-1 matrix A avoids a pattern P given as a 0-1 matrix we mean that no submatrix of A either equals P or can be transformed into P by replacing some 1 entries with 0 entries. We present a new method for estimating the maximal number of the 1 entries in a matrix that avoids a certain pattern. Applying this method we give a linear bound on the maximal number of the 1 entries in an n by n matrix avoiding pattern L1 and thereby we answer the question that was asked by Gábor Tardos. Furthermore, we use our approach on patterns related to L1. 相似文献
19.
Bendong Lou 《Proceedings of the American Mathematical Society》1999,127(8):2259-2264
This paper investigates the fixed points for self-maps of a closed set in a space of abstract continuous functions. Our main results essentially extend the Banach contracting mapping principle. An application to integro-differential equations is given.
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M. Chara 《Journal of Pure and Applied Algebra》2011,215(11):2603-2614
In this paper we prove results on the number of rational places in extensions of Kummer type over finite fields and give sufficient conditions for non-trivial lower bounds on the number of rational places at each step of sequences of function fields over a finite field, that we call (a, b)-sequences. In the case of a prime field, we apply these results to the study of rational places in certain sequences of function fields of Kummer type. 相似文献