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We examine the coefficients in front of Chern numbers for complex genera, and pay special attention to the Td 1 2 -genus, the Γ-genus as well as the Todd genus. Some related geometric applications to hyper-Kähler and Calabi-Yau manifolds are discussed. Along this line and building on the work of Doubilet in 1970s, various Hoffman-type formulas for multiple-(star) zeta values and transition matrices among canonical bases of the ring of symmetric functions can be uniformly treated in a more general framework. [ABSTRACT FROM AUTHOR]

In this paper, we extend the chromatic symmetric function X to a chromatic k-multisymmetric function X k , defined for graphs equipped with a partition of their vertex set into k parts. We demonstrate that this new function retains the basic properties and basis expansions of X , and we give a method for systematically deriving new linear relationships for X from previous ones by passing them through X k. In particular, we show how to take advantage of homogeneous sets of G (those S ⊆ V (G) such that each vertex of V (G) ﹨ S is either adjacent to all of S or is nonadjacent to all of S) to relate the chromatic symmetric function of G to those of simpler graphs. Furthermore, we show how extending this idea to homogeneous pairs S 1 ⊔ S 2 ⊆ V (G) generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs. [ABSTRACT FROM AUTHOR]

We introduce a family of varieties Y n , λ , s , which we call the Δ -Springer varieties , that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring H ⁎ (Y n , λ , s) and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induction products of Specht modules with trivial modules. The λ = (1 k) case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at t = 0. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety Y n , λ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud–Saltman rank variety and diagonal matrices. [ABSTRACT FROM AUTHOR]

A Boolean function f : { 0 , 1 } n → { 0 , 1 } is weighted symmetric if there exist a function g : Z → { 0 , 1 } and integers w 0 , w 1 , ... , w n such that f (x 1 , ... , x n) = g (w 0 + ∑ i = 1 n w i x i) holds. In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2 n even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. As a special case, we obtain an algorithm for the maximum satisfiability problem that runs in time poly (n t) ⋅ 2 n − n 1 / O (t) for instances with n variables and O (n t) clauses. Through the analysis of our algorithms, we show average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits. [ABSTRACT FROM AUTHOR]

Abstract We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiples of the odd prime p the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pairs of partitions. As a result, for each partition λ , we discovered the number of compositions δ such that δ can be rearranged to λ and whose partial sums of δ are not divisible by p. [ABSTRACT FROM AUTHOR]

An ( r , k ) -parking function of length n may be defined as a sequence ( a 1 , … , a n ) of positive integers whose increasing rearrangement b 1 ≤ ⋯ ≤ b n satisfies b i ≤ k + ( i − 1 ) r . The case r = k = 1 corresponds to ordinary parking functions. We develop numerous properties of ( r , k ) -parking functions. In particular, if F n ( r , k ) denotes the Frobenius characteristic of the action of the symmetric group S n on the set of all ( r , k ) -parking functions of length n , then we find a combinatorial interpretation of the coefficients of the power series ( ∑ n ≥ 0 F n ( r , 1 ) t n ) k for any k ∈ Z . When k > 0 , this power series is just ∑ n ≥ 0 F n ( r , k ) t n ; when k < 0 , we obtain a dual to ( r , k ) -parking functions. We also give a q -analogue of this result. For fixed r , we can use the symmetric functions F n ( r , 1 ) to define a multiplicative basis for the ring Λ of symmetric functions. We investigate some of the properties of this basis. [ABSTRACT FROM AUTHOR]

The symmetric group S n acts on the polynomial ring Q [ x n ] = Q [ x 1 , … , x n ] by variable permutation. The invariant ideal I n is the ideal generated by all S n -invariant polynomials with vanishing constant term. The quotient R n = Q [ x n ] I n is called the coinvariant algebra . The coinvariant algebra R n has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization I n , k ⊆ Q [ x n ] of the ideal I n indexed by two positive integers k ≤ n . The corresponding quotient R n , k : = Q [ x n ] I n , k carries a graded action of S n and specializes to R n when k = n . We generalize many of the nice properties of R n to R n , k . In particular, we describe the Hilbert series of R n , k , give extensions of the Artin and Garsia–Stanton monomial bases of R n to R n , k , determine the reduced Gröbner basis for I n , k with respect to the lexicographic monomial order, and describe the graded Frobenius series of R n , k . Just as the combinatorics of R n are controlled by permutations in S n , we will show that the combinatorics of R n , k are controlled by ordered set partitions of { 1 , 2 , … , n } with k blocks. The Delta Conjecture of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of R n , k is (up to a minor twist) the t = 0 specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded S n -module V n , k whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module R n , k solves this problem in the specialization t = 0 . [ABSTRACT FROM AUTHOR]

The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson and Haglund–Remmel–Wilson which give equidistribution results for statistics related to inversion count and major index on objects related to ordered set partitions. Our results generalize the famous result of MacMahon that major index and inversion number share the same distribution on permutations. [ABSTRACT FROM AUTHOR]

The main aim of the article is to prove that the symmetric function Φ n ( x , r ) = ∏ i 1 + i 2 + ⋯ + i n = r ( x 1 i 1 + x 2 i 2 + ⋯ + x n i n ) is Schur geometrically convex for x ∈ R + + n and fixed r ∈ N + = { 1 , 2 , ⋯ } , where i 1 , i 2 , ⋯ , i n are non-negative integers. Further, we obtain Φ n ( x , r ) is also Schur m -power convex for m ≤ 0 . As applications, a Klamkin–Newman type inequality is derived. Finally, we list a counter example to illustrate Φ n ( x , r ) is neither Schur convex nor Schur concave. [ABSTRACT FROM AUTHOR]

We relate the m -truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when m = 2 . Each g -vector cone G ◇ l of these cluster algebras controls the 2-truncated Kronecker products for all symmetric functions of degree no greater than l . As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all G ◇ l 's. As an application, we compute some invariant rings. [ABSTRACT FROM AUTHOR]

The classical Littlewood–Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood–Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood–Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood–Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van Willigenburg. [ABSTRACT FROM AUTHOR]

In 2004 Rosas and Sagan asked whether there was a way to define a basis in the algebra of symmetric functions in noncommuting variables, NCSym, having properties analogous to the classical Schur functions. This was because they had constructed a partial such set that was not a basis. We answer their question by defining Schur functions in noncommuting variables using a noncommutative analogue of the Jacobi-Trudi determinant. Our Schur functions in NCSym map to classical Schur functions under commutation, and a subset of them indexed by set partitions forms a basis for NCSym. Amongst other properties, Schur functions in NCSym also satisfy a noncommutative analogue of the product rule for classical Schur functions in terms of skew Schur functions. We also show how Schur functions in NCSym are related to Specht modules, and naturally refine the Rosas-Sagan Schur functions. Moreover, by generalizing Rosas-Sagan Schur functions to skew Schur functions in the natural way, we prove noncommutative analogues of the Littlewood-Richardson rule and coproduct rule for them. Finally, we relate our functions to noncommutative symmetric functions by proving a subset of our functions are natural extensions of noncommutative ribbon Schur functions, and immaculate functions indexed by integer partitions. [ABSTRACT FROM AUTHOR]

Abstract: In the study of Zeilbergerʼs conjecture on an integer sequence related to the Catalan numbers, Lassalle proposed the following conjecture. Let denote the rising factorial, and let denote the algebra of symmetric functions with real coefficients. If φ is the homomorphism from to defined by for some , then for any Schur function , the value is positive. In this paper, we provide an affirmative answer to Lassalleʼs conjecture by using the Laguerre–Pólya–Schur theory of multiplier sequences. [Copyright &y& Elsevier]

Abstract: We propose a method for diagonalizing matrices with entries in commutative rings. The point of departure is to split the characteristic polynomial of the matrix over a (universal) splitting algebra, and to use the resulting universal roots to construct eigenvectors of the matrix. A crucial point is to determine when the determinant of the eigenvector matrix, that is the matrix whose columns are the eigenvectors, is regular in the splitting algebra. We show that this holds when the matrix is generic, that is, the entries are algebraically independent over the base ring. It would have been desirable to have an explicit formula for the determinant in the generic case. However, we have to settle for such a formula in a special case that is general enough for proving regularity in the general case. We illustrate the uses of our results by proving the Spectral Mapping Theorem, and by generalizing a fundamental result from classical invariant theory. [Copyright &y& Elsevier]

Abstract: Recently a new basis for the Hopf algebra of quasisymmetric functions QSym, called quasisymmetric Schur functions, has been introduced by Haglund, Luoto, Mason, van Willigenburg. In this paper we extend the definition of quasisymmetric Schur functions to introduce skew quasisymmetric Schur functions. These functions include both classical skew Schur functions and quasisymmetric Schur functions as examples, and give rise to a new poset that is analogous to Young''s lattice. We also introduce a new basis for the Hopf algebra of noncommutative symmetric functions NSym. This basis of NSym is dual to the basis of quasisymmetric Schur functions and its elements are the pre-image of the Schur functions under the forgetful map . We prove that the multiplicative structure constants of the noncommutative Schur functions, equivalently the coefficients of the skew quasisymmetric Schur functions when expanded in the quasisymmetric Schur basis, are nonnegative integers, satisfying a Littlewood–Richardson rule analogue that reduces to the classical Littlewood–Richardson rule under χ. As an application we show that the morphism of algebras from the algebra of Poirier–Reutenauer to Sym factors through NSym. We also extend the definition of Schur functions in noncommuting variables of Rosas–Sagan in the algebra NCSym to define quasisymmetric Schur functions in the algebra NCQSym. We prove these latter functions refine the former and their properties, and project onto quasisymmetric Schur functions under the forgetful map. Lastly, we show that by suitably labeling , skew quasisymmetric Schur functions arise in the theory of Pieri operators on posets. [Copyright &y& Elsevier]

Abstract: We give a combinatorial realization of extremal weight crystals over the quantum group of type and their Littlewood–Richardson rule. Based on this description, we show that the Grothendieck ring generated by the isomorphism classes of extremal weight -crystals is isomorphic to the Weyl algebra of infinite rank, and hence each isomorphism class is realized as a differential operator or non-commutative Schur function acting on the algebra of symmetric functions. We also find a duality between extremal weight -crystals and generalized Verma -crystals appearing in the crystal of the Fock space with infinite level, which recovers the generalized Cauchy identity for Schur operators in a bijective and crystal theoretic way. [Copyright &y& Elsevier]

Abstract: The “carries” when n random numbers are added base b form a Markov chain with an “amazing” transition matrix determined in a 1997 paper of Holte. This same Markov chain occurs in following the number of descents when n cards are repeatedly riffle shuffled. We give generating and symmetric function proofs and determine the rate of convergence of this Markov chain to stationarity. Similar results are given for type B shuffles. We also develop connections with Gaussian autoregressive processes and the Veronese mapping of commutative algebra. [Copyright &y& Elsevier]

Abstract: New sufficient conditions and necessary conditions are developed for two skew diagrams to give rise to the same skew Schur function. The sufficient conditions come from a variety of new operations related to ribbons (also known as border strips or rim hooks). The necessary conditions relate to the extent of overlap among the rows or among the columns of the skew diagram. [Copyright &y& Elsevier]

Abstract: We use the theory of symmetric functions to enumerate various classes of alternating permutations w of . These classes include the following: (1) both w and are alternating, (2) w has certain special shapes, such as , under the RSK algorithm, (3) w has a specified cycle type, and (4) w has a specified number of fixed points. We also enumerate alternating permutations of a multiset. Most of our formulas are umbral expressions where after expanding the expression in powers of a variable E, is interpreted as the Euler number . As a small corollary, we obtain a combinatorial interpretation of the coefficients of an asymptotic expansion appearing in Ramanujan''s “Lost” Notebook. [Copyright &y& Elsevier]

Abstract: In this paper, which is a continuation of [V. Timofte, On the positivity of symmetric polynomial functions. Part I: General results, J. Math. Anal. Appl. 284 (2003) 174–190] and [V. Timofte, On the positivity of symmetric polynomial functions. Part II: Lattice general results and positivity criteria for degrees 4 and 5, J. Math. Anal. Appl., in press], we study properties of extremal polynomials of degree 4, and we give the construction of some of them. The main results are Theorems 9, 13, 15, 16, and 18. [Copyright &y& Elsevier]

Abstract: The notion of a -algebra has a non-linear analogue, whose purpose it is to control operations on commutative rings rather than linear operations on abelian groups. These plethories can also be considered non-linear generalizations of cocommutative bialgebras. We establish a number of category-theoretic facts about plethories and their actions, including a Tannaka–Krein-style reconstruction theorem. We show that the classical ring of Witt vectors, with all its concomitant structure, can be understood in a formula-free way in terms of a plethystic version of an affine blow-up applied to the plethory generated by the Frobenius map. We also discuss the linear and infinitesimal structure of plethories and explain how this gives Bloch''s Frobenius operator on the de Rham–Witt complex. [Copyright &y& Elsevier]

Letting τ denote the inverse transpose automorphism of GL(n,q), a formula is obtained for the number of g in GL(n,q) so that ggτ is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, ggτ behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL(n,q) and of symmetric function theory, including a new identity for Hall–Littlewood polynomials. We obtain information about random elements of finite symplectic groups in even characteristic, and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of GL(n,q) generated by the inverse transpose automorphism. We give a second approach to these results using the theory of bilinear forms over a field. The results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups, and we give a few examples in this direction. [Copyright &y& Elsevier]

We prove that a real symmetric polynomial inequality of degree d&ges;2 holds on R+n if and only if it holds for elements with at most ⌊d/2⌋ distinct non-zero components, which may have multiplicities. We establish this result by solving a Cauchy problem for ordinary differential equations involving the symmetric power sums; this implies the existence of a special kind of paths in the minimizer of some restriction of the considered polynomial function. In the final section, extensions of our results to the whole space Rn are outlined. The main results are Theorems 5.1 and 5.2 with Corollaries 2.1 and 5.2, and the corresponding results for Rn from the last subsection. Part II will contain a discussion on the ordered vector space Hd[n] in general, as well as on the particular cases of degrees d=4 and d=5 (finite test sets for positivity in the homogeneous case and other sufficient criteria). [Copyright &y& Elsevier]

The study of asymptotic properties of the conjugacy class of a random element of the finite affine group leads one to define a probability measure on the set of all partitions of all positive integers. Four different probabilistic understandings of this measure are given—three using symmetric function theory and one using Markov chains. This leads to non-trivial enumerative results. Cycle index generating functions are derived and are used to compute the large dimension limiting probabilities that an element of the affine group is separable, cyclic, or semisimple and to study the convergence to these limits. The semisimple limit involves both Rogers–Ramanujan identities. This yields the first examples of such computations for a maximal parabolic subgroup of a finite classical group. [Copyright &y& Elsevier]

By combining the Gessel–Xin method with plethystic substitutions, we obtain a recursion for a symmetric function generalization of the q -Dyson constant term identity also known as the Zeilberger–Bressoud q -Dyson theorem. This yields a constant term identity which generalizes the non-zero part of Kadell's orthogonality ex-conjecture and a result of Károlyi, Lascoux and Warnaar. [ABSTRACT FROM AUTHOR]

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the q -Dyson constant term identity or the Zeilberger–Bressoud q -Dyson theorem. The non-zero part of Kadell's orthogonality conjecture is a constant term identity indexed by a weak composition v = (v 1 , ... , v n) in the case when only one v i ≠ 0. This conjecture was first proved by Károlyi, Lascoux and Warnaar in 2015. They further formulated a closed-form expression for the above mentioned constant term in the case when all the parts of v are distinct. Recently we obtained a recursion for this constant term provided that the largest part of v occurs with multiplicity one in v. In this paper, we generalize our previous result to all weak compositions v. [ABSTRACT FROM AUTHOR]

We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specialises to (extended) k -deletion, and we give two methods to obtain numerous new bases from weighted graphs for the algebra of symmetric functions. Moreover, we classify when two weighted paths have equal extended chromatic symmetric functions by proving this is equivalent to the classification of equal ribbon Schur functions. This latter classification is dependent on the operation composition of compositions, which we generalise to composition of graphs. We then apply our generalisation to obtain infinitely many families of weighted graphs whose members have equal extended chromatic symmetric functions. [ABSTRACT FROM AUTHOR]

For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length statistics. This involves a variety of tools that are not standard in the subject, such as algebraic combinatorics (Schur polynomials), probability theory (weak convergence of measures, characteristic functions), and harmonic analysis (Fourier transforms of distributions). We provide instructive examples which demonstrate the power and generality of our techniques. We also highlight unexpected consequences in the theory of homogeneous symmetric functions. [ABSTRACT FROM AUTHOR]

For any finite group G it is an interesting question to ask which ordinary irreducible representations of G remain irreducible in a given characteristic p. We answer this question for p = 2 when G is the proper double cover of the alternating group. As a key ingredient in the proof, we prove a formula for the decomposition numbers in Rouquier blocks of double covers of symmetric groups, in terms of Schur P-functions. [ABSTRACT FROM AUTHOR]

Hypergeometric functions and zonal polynomials are the tools usually addressed in the literature to deal with the expected value of the elementary symmetric functions in non-central Wishart latent roots. The method here proposed recovers the expected value of these symmetric functions by using the umbral operator applied to the trace of suitable polynomial matrices and their cumulants. The employment of a suitable linear operator in place of hypergeometric functions and zonal polynomials was conjectured by de Waal in (1972). Here we show how the umbral operator accomplishes this task and consequently represents an alternative tool to deal with these symmetric functions. When special formal variables are plugged in the variables, the evaluation through the umbral operator deletes all the monomials in the latent roots except those contributing in the elementary symmetric functions. Cumulants further simplify the computations taking advantage of the convolution structure of the polynomial trace. Open problems are addressed at the end of the paper. [ABSTRACT FROM AUTHOR]