Hook,line and sinker: A bijective proof of the skew shifted hook-length formula

A few years ago, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes, both straight and shifted. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall.Recently, the formula for skew straight shapes was proved by the author via a simple bumping algorithm. The aim of this paper is to extend this result to skew shifted shapes. Since straight skew shapes are special cases of skew shifted shapes, this is a bijection that proves the whole family of hook-length formulas, and is also the simplest known bijective proof for shifted (non-skew) shapes. The complexity of the algorithm is studied, and a weighted generalization of Naruse’s formula is also presented.     

A generalization of the symmetry between complete and elementary symmetric functions

A generalization for the symmetry between complete symmetric functions and elementary symmetric functions is given. As corollaries we derive the inverse of a triangular Toeplitz matrix and the expression of the Toeplitz-Hessenberg determinant. A very large variety of identities involving integer partitions and multinomial coefficients can be generated using this generalization. The partitioned binomial theorem and a new formula for the partition function p(n) are obtained in this way.     

Probability density functions of some skew tent maps

We consider a family of chaotic skew tent maps. The skew tent map is a two-parameter, piecewise-linear, weakly-unimodal, map of the interval F_a,b. We show that F_a,b is Markov for a dense set of parameters in the chaotic region, and we exactly find the probability density function (pdf), for any of these maps. It is well known (Boyarsky A, Góra P. Laws of chaos: invariant measures and dynamical systems in one dimension. Boston: Birkhauser, 1997), that when a sequence of transformations has a uniform limit F, and the corresponding sequence of invariant pdfs has a weak limit, then that invariant pdf must be F invariant. However, we show in the case of a family of skew tent maps that not only does a suitable sequence of convergent sequence exist, but they can be constructed entirely within the family of skew tent maps. Furthermore, such a sequence can be found amongst the set of Markov transformations, for which pdfs are easily and exactly calculated. We then apply these results to exactly integrate Lyapunov exponents.     

Double circulant matrices

Double circulant matrices are introduced and studied. By a matrix-theoretic method, the rank r of a double circulant matrix is computed, and it is shown that any consecutive r rows of the double circulant matrix are linearly independent. As a generalization, multiple circulant matrices are also introduced. Two questions on square double circulant matrices are posed.     

Extension of a quadratic transformation due to Exton

By applying various known summation theorems to a general formula based upon Bailey’s transform theorem due to Slater, Exton has obtained numerous new quadratic transformations involving hypergeometric functions of two and of higher order. Some of the results have typographical errors and have been corrected recently by Choi and Rathie. In addition, two new quadratic transformation formulæ were also obtained [Junesang Choi, A.K. Rathie, Quadratic transformations involving hypergeometric functions of two and higher order, EAMJ, East Asian Math. J. 22 (2006) 71-77]. The aim of this research paper is to obtain a generalization of one of the Exton’s quadratic transformation. The results are derived with the help of generalized Kummer’s theorem obtained earlier by Lavoie, Grondin and Rathie. As special cases, we mention six interesting results closely related to that of Exton’s result.     

Transformations of Border Strips and Schur Function Determinants

We introduce the notion of the cutting strip of an outside decomposition of a skew shape, and show that cutting strips are in one-to-one correspondence with outside decompositions for a given skew shape. Outside decompositions are introduced by Hamel and Goulden and are used to give an identity for the skew Schur function that unifies the determinantal expressions for the skew Schur functions including the Jacobi-Trudi determinant, its dual, the Giambelli determinant and the rim ribbon determinant due to Lascoux and Pragacz. Using cutting strips, one obtains a formula for the number of outside decompositions of a given skew shape. Moreover, one can define the basic transformations which we call the twist transformation among cutting strips, and derive a transformation theorem for the determinantal formula of Hamel and Goulden. The special case of the transformation theorem for the Giambelli identity and the rim ribbon identity was obtained by Lascoux and Pragacz. Our transformation theorem also applies to the supersymmetric skew Schur function.     

Kibble–Slepian formula and generating functions for 2D polynomials

We prove a generalization of the Kibble–Slepian formula (for Hermite polynomials) and its unitary analogue involving the 2D Hermite polynomials recently proved in [16]. We derive integral representations for the 2D Hermite polynomials which are of independent interest. Several new generating functions for 2D q-Hermite polynomials will also be given.     

The automorphism-order in finite groups

The notion of automorphism-order is introduced as a generalization of elemental order in finite groups. Some theorems involving orders of elements are then generalized. Divisibility properties involving this concept are considered. Necessary and sufficient conditions for an abelian group to be represented by number-theoretic functions involving divisibility properties are given. Explicit formulas of these functions are also given.     

A generalization of the Gauss hypergeometric series

A generalization of the Gauss hypergeometric function to t variables is given, and the Euler identity is shown to hold for this generalized function. The corresponding generalization of the Saalschütz theorem is also obtained.     

Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables

In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields are isomorphic as algebras to a multivariate skew polynomial ring whose ring morphism from the field to the ring of square matrices is diagonal, and whose derivation is the zero derivation. Furthermore, we prove that two such representations only differ in a permutation on the field automorphisms appearing in the corresponding diagonal. The algebra isomorphisms are given by affine transformations of variables and preserve evaluations and degrees. In addition, ours proofs show that the simplified form of multivariate skew polynomial rings can be found computationally and explicitly.     

The inverse of circulant matrix

A direct method is proposed to get the inverse matrix of circulant matrix that find important application in engineering, the elements of the inverse matrix are functions of zero points of the characteristic polynomial g(z) and g′(z) of circulant matrix, four examples to get the inverse matrix are presented in the paper.     

Generalized skew bisubmodularity: A characterization and a min–max theorem

Huber, Krokhin, and Powell (2013) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain. In this paper we consider a natural generalization of the concept of skew bisubmodularity and show a connection between the generalized skew bisubmodularity and a convex extension over rectangles. We also analyze the dual polyhedra, called skew bisubmodular polyhedra, associated with generalized skew bisubmodular functions and derive a min–max theorem that characterizes the minimum value of a generalized skew bisubmodular function in terms of a minimum-norm point in the associated skew bisubmodular polyhedron.     

Minimal rank of abelian group matrices

The minimal rank of abelian group matrices with positive integral entries is determined.The corresponding problem for circulant matrices have been investigated by Ingleton and more recently by Shiu-Ma-Fang. Our work can be viewed as a generalization of their results, since a group matrix becomes circulant when the group is cyclic.     

Skew Hadamard designs and their codes

Skew Hadamard designs (4n – 1, 2n – 1, n – 1) are associated to order 4n skew Hadamard matrices in the natural way. We study the codes spanned by their incidence matrices A and by I + A and show that they are self-dual after extension (resp. extension and augmentation) over fields of characteristic dividing n. Quadratic Residues codes are obtained in the case of the Paley matrix. Results on the p-rank of skew Hadamard designs are rederived in that way. Codes from skew Hadamard designs are classified. An optimal self-dual code over GF(5) is rediscovered in length 20. Six new inequivalent [56, 28, 16] self-dual codes over GF(7) are obtained from skew Hadamard matrices of order 56, improving the only known quadratic double circulant code of length 56 over GF(7).     

从p—m链到随机环境中的马氏链

第一节引进了p一m链的概念,并用之构造了与它相应的随机环境中的马氏链和绕积马氏链、第二节引进了一系列与随机环境中的马氏链相关的概率特性函数,并得到了这些函数之间的一系列关系.这些结果是经典马氏链的相应结果的一般化,它们在随机环境中的马氏链的极限理论的研究中是很有用的。     

Certain weighted averages of generalized Ramanujan sums

We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Tóth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose a generalization of Ramanujan sum given by E. Cohen and derive the weighted averages corresponding to the versions of the weighted averages established by Tóth.     

Generalized inverses of circulant and generalized circulant matrices

An expression for the Moore-Penrose inverse of certain singular circulants by S.R. Searle is generalized to include all circulants. Similar expressions are given for the Moore-Penrose inverse of block circulants with circulant blocks, level-q circulants, k-circulants where |k|=1, and certain other matrices which are the product of a permutation matrix and a circulant. Expressions for other generalized inverses are given.     

Cylindric skew Schur functions

Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining these motivations, we study cylindric skew Schur functions from the point of view of Schur-positivity. Using a result of I. Gessel and C. Krattenthaler, we generalise a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus giving an expansion of an arbitrary cylindric skew Schur function in terms of skew Schur functions. While we show that no non-trivial cylindric skew Schur functions are Schur-positive, we conjecture that this can be reconciled using the new concept of cylindric Schur-positivity.     

On the set covering polyhedron of circulant matrices

A well known family of minimally nonideal matrices is the family of the incidence matrices of chordless odd cycles. A natural generalization of these matrices is given by the family of circulant matrices. Ideal and minimally nonideal circulant matrices have been completely identified by Cornuéjols and Novick [G. Cornuéjols, B. Novick, Ideal 0 - 1 matrices, Journal of Combinatorial Theory B 60 (1994) 145–157]. In this work we classify circulant matrices and their blockers in terms of the inequalities involved in their set covering polyhedra. We exploit the results due to Cornuéjols and Novick in the above-cited reference for describing the set covering polyhedron of blockers of circulant matrices. Finally, we point out that the results found on circulant matrices and their blockers present a remarkable analogy with a similar analysis of webs and antiwebs due to Pêcher and Wagler [A. Pêcher, A. Wagler, A construction for non-rank facets of stable set polytopes of webs, European Journal of Combinatorics 27 (2006) 1172–1185; A. Pêcher, A. Wagler, Almost all webs are not rank-perfect, Mathematical Programming Series B 105 (2006) 311–328] and Wagler [A. Wagler, Relaxing perfectness: Which graphs are ‘Almost’ perfect?, in: M. Groetschel (Ed.), The Sharpest Cut, Impact of Manfred Padberg and his work, in: SIAM/MPS Series on Optimization, vol. 4, Philadelphia, 2004; A. Wagler, Antiwebs are rank-perfect, 4OR 2 (2004) 149–152].     

A Two Function Metaminimax Theorem

A generalization of Simons's metaminimax theorem to a metaminimax theorem involving two functions is given.