commutant structure of u n 1 u transformations in this paper we have found irreducible representations irreps of the algebra of partially transposed permutation operators on the last subsystem we give here the direct method of irreps construction for an arbitrary number of subsystems n and local dimension d our method is inspired by the representation theory of the symmetric group s n the theory of brauer algebras and walled brauer algebras

irreducible representations of brauer algebras irreducible representations of brauer algebras are constructed by using the induced representation and the linear equation method as examples some matrix representations of brauer algebras df n with f or 5 are presented

on the blocks of the walled brauer algebra we determine the blocks of the walled brauer algebra in characteristic zero these can be described in terms of orbits of the action of a weyl group of type a on a certain set of weights in positive characteristic we give a linkage principle in terms of orbits of the corresponding affine weyl group we also classify the semisimple walled brauer algebras in all characteristics

a machine for producing inequalities involving immanants and other generalized matrix functions abstract there are many known inequalities involving the restriction of immanants and other generalized matrix functions to the set of positive semidefinite hermitian matrices we present a general scheme for obtaining such inequalities matrix theoretic refinements of lieb s and fischer s inequalities recently discovered by pate are obtainable using the scheme other known inequalities are discussed and we derive new inequalities that relate characters constructed from generalized diagrams

maximization of a matrix function related to the dittert conjecture abstract let k n denote the set of all nonnegative n n matrices whose entries have sum n and let j n 1 n n n for k 1 n and for a k n with row sum s r 1 r n and column sums c 1 c n let k be defined by k a q k i r i j c j per a we propose a problem of maximizing k of which the dittert conjecture is a special case and obtain some results related to this problem many of known theorems for n are shown to hold for k k 1 n

permanental bounds for nonnegative matrices via decomposition abstract we investigate an old suggestion of a e brouwer we call decomposition for constructing a class of permanental upper bounds for nonnegative matrices a from a single permanental upper bound u b for 0 1 matrices b for certain feasible u which include the minc bregman bound u b m b and the jurkat ryser bound u b j b we can identify the best and worst of these decomposition bounds the best decomposition bound the star bound u a is the only decomposition bound which agrees with u on the 0 1 matrices if u j then u a turns out to be the very bound uj a used by jurkat and ryser to obtain j as a special case if u m then u a is a new upper bound um a we believe its sharpened version u g m a to be the best extant permanental upper bound for nonnegative matrices as well as for 0 1 matrices

minimum permanents of multiplexes abstract we consider the minimum value of the permanent over certain faces of n the polytope of n n doubly stochastic matrices the faces we consider are determined by two types of bipartite graphs a complex and a multiplex which is a collection of complexes glued together in a particular way we show that the minimum value of the permanent over a face determined by a complex or multiplex is at least 1 2 n 1 and that the minimum need not be at a barycentric matrix we also verify that the tie point conjecture is true for complexes

immanants of combinatorial matrices abstract the arbitrary immanants of three matrices whose determinants are known to be generating functions for sets of combinatorial objects are examined combinatorial interpretations are given for the immanants of the matrix tree matrix and a special case of the jacobi trudi matrix these allow us to deduce immediately the nonnegativity of the coefficients in the expansion of the immanants a conjecture is made about the nonnegativity of coefficients of the expansion of the immanant of the jacobi trudi matrix in the general case this nonnegativity result is seen to fail for the hankel matrix and combinatorial reasons for this failure are given all results can be translated into statements about the nonnegativity of schur function expansions for the related symmetric functions

a note on the analogue of oppenheim s inequality for permanents the inequality per a b pera perb is verified for nonnegative 2 2 and 3 3 hermitian matrices where a b is the hadamard product ofa and b

an approach to the permanental dominance conjecture abstract the permanental dominance conjecture for positive semidefinite hermitian matrices a has attracted much interest in the last ten years a stronger conjecture that the maximal eigenvalue of the schur power matrix a is per a would if true imply the dominance of the permanent we prove that should the maximum eigenvalue conjecture fail in the real case then the smallest n for which it fails must be such that it fails at a singular matrix having certain properties including zero row sums many of our results were also obtained independently by j p holmes and tom pate at auburn university

distillation of entanglement by projection on permutationally invariant subspaces we consider distillation of entanglement from two qubit states which are mixtures of two pure entangled states and one pure product state which is orthogonal to them we distill entanglement from such states by projecting n copies of the state on a permutationally invariant subspace and then applying one way hashing protocol we find analytical expressions for the rate of the protocol and show that for wide range of parameters this protocol achieves higher rates than previous ones we also generalize this method to higher dimensional systems to get analytical expression for two qubit case we faced a mathematical problem of diagonalizing a family of matrices enjoying some symmetries w r t the symmetric group we have solved this problem in two ways i directly by the use of schur weyl decomposition and young symmetrizers and ii showing that the problem is equivalent to a problem of diagonalizing adjacency matrices in a particular instance of a so called algebraic association scheme

on the decomposition of tensor products of the representations of the classical groups by means of the universal characters on definit les caracteres universels d une representation rationnelle de gl n c et on donne les formules de decomposition des produits tensoriels entre eux en termes de coefficients de littlewood richardson on demontre les formules de decomposition du produit tensoriel de deux representations irreductibles de sp 2n c et de so n c

a hadamard inequality for the second immanant abstract denote by h n the convex cone of n by n positive semi definite hermitian matrices let d2 be the normalized immanant afforded by sn and the partition 2 1n 2 then h a 1 det a d2 a a hn where h a is the main diagonal product of a and is approximately n 1 e

inequalities and identities for generalized matrix functions abstract let be a character on the symmetric group s n and let a a ij be an n by n matrix the function d a s n n t 1 a t t is called a generalized matrix function if is an irreducible character then d is called an immanent for example if is the alternating character then d is the determinant and if 1 then d is called the permanent denoted per suppose that a is positive semidefinite hermitian we prove that the inequality 1 id d a per a holds for a variety of characters including the irreducible ones corresponding to the partitions n 1 1 and n 2 1 1 of n the main technique used to prove these inequalities is to express the immanents as sums of products of principal subpermanents these expressions for the immanents come from analogous expressions for schur polynomials by means of a correspondence of d e littlewood

kazhdan lusztig immanants and products of matrix minors abstract using the kazhdan lusztig basis c w 1 w s n for the symmetric group algebra we obtain nonnegativity properties of certain polynomials in matrix minors in particular we show that the application of these polynomials to jacobi trudi matrices yields symmetric functions which are equal to nonnegative linear combinations of schur functions

an extremal property of the permanent and the determinant abstract given an n n matrix a define the n n matrix a with rows and columns indexed by the permutation group s n with entry n i 1 a i i it is shown that if a is positive semidefinite then det a is the smallest eigenvalue of a it is conjectured that per a is the largest eigenvalue of a and the conjecture proved for n 3 several known and some unknown inequalities are derived as consequences

galois connections and tense operators on q effect algebras for effect algebras the so called tense operators were already introduced by chajda and paseka they presented also a canonical construction of them using the notion of a time frame tense operators express the quantifiers it is always going to be the case that and it has always been the case that and hence enable us to express the dimension of time both in the logic of quantum mechanics and in the many valued logic a crucial problem concerning tense operators is their representation having an effect algebra with tense operators we can ask if there exists a time frame such that each of these operators can be obtained by the canonical construction to approximate physical real systems as best as possible we introduce the notion of a q effect algebra and we solve this problem for q tense operators on q representable q jauch piron q effect algebras

inequalities involving immanants abstract we summarize the recent progress toward determining the ordering of the normalized immanants regarded as generalized matrix functions and restricted to the n n positive semidefinite hermitian matrices included are the most recent results of heyfron and pate moreover we describe some of the techniques used

structure and properties of the algebra of partially transposed permutation operators we consider the structure of algebra of operators acting in n fold tensor product space which are partially transposed on the last term using purely algebraical methods we show that this algebra is semi simple and then considering its regular representation we derive basic properties of the algebra in particular we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations it appears that there are two kinds of irreducible representations of the algebra the first one is strictly connected with the representations of the group s n 1 induced by irreducible representations of the group s n 2 the second kind is structurally connected with irreducible representations of the group s n 1

on a conjecture of e dittert abstract let kn denote the set of all n x n nonnegative matrices whose entries have sum n and let be a real valued function defined on kn by x in 1 n j 1cj n per x for x e kn with row sum vector r1 rn and column sum vector cl hellip cn for the same x let ij x k irk 1 jc1 per x i j a kn is called a maximizing matrix if a x for all x kn dittert s conjecture asserts that jn 1 n n n is the unique maximizing matrix on kn in this paper the following are proved i if a aij is a maximizing matrix on kn then ij a a if aij 0 and ij a a if aij 0 ii the conjecture is true for n 3

tensor inequalities ξ functions and inequalities involving immanants abstract if f is a complex valued function with domain s n the symmetric group on 1 2 n then we define the matrix function df in the usual way if is a partition of n and denotes the associated irreducible character of s n then d which we abbreviate to d is known as an immanant the permanent dominance conjecture of lieb specialized to the symmetric groups asserts that if is a partition of n then d a deg per a for each a h n the set of n n positive semi definite hermitian matrices recently it was shown t h pate proc london math soc 76 2 1998 307 358 that if is a partition of n of the form p q r 2s 1t then d a deg per a for each a h n we improve upon this result by showing that this same inequality holds for all a h n when is a partition of the form p qw r 2s 1t where 0 w 2 thus permanent dominance holds for all immanants whose associated partitions are of the form p q2 r we also show that permanent dominance holds for all immanants whose associated partitions are of the form n p nk as long as n is sufficiently large inequalities involving immanants have often been obtained using a special kind of class function known as a function our results on immanants are obtained by analyzing a new set of class functions called functions that are more fundamental than the functions in the sense that each function is a non negative linear combination of functions the functions arise from a study of tensor contractions and a special collection of quadratic forms defined on spaces of bi symmetric tensors

an update on minc s survey of open problems involving permanents abstract we summarise the progress which has been made since 1986 on the conjectures and open problems listed in h minc s survey articles on the theory of permanents

pseudocomplemented lattice effect algebras and existence of states we prove that in every pseudocomplemented atomic lattice effect algebra the subset of all pseudocomplements is a boolean algebra including the set of sharp elements as a subalgebra as an application we show families of effect algebras for which the existence of a pseudocomplementation implies the existence of states these states can be obtained by smearing of states existing on the boolean algebra of sharp elements

approximating the permanent of graphs with large factors let g u v e be a bipartite graph with u v n the factor size of g f is the maximum number of edge disjoint perfect matchings in g we characterize the complexity of counting the number of perfect matchings in classes of graphs parameterized by factor size we describe the simple algorithm which is an approximation algorithm for the permanent that is a natural simplification of the algorithm suggested by broder 1986 and analyzed by jerrum and sinclair 1988a b compared to the algorithm by jerrum and sinclair 1988a b the simple algorithm achieves a polynomial speed up in the running time to compute the permanent a combinatorial lemma is used to prove that the simple algorithm runs in time no n f thus 1 for all constants 0 the simple algorithm runs in polynomial time for graphs with factor size at least n 2 for some constant c the simple algorithm is the fastest known approximation for graphs with factor size at least c log n compare with the approximation algorithms described in karmarkar 1988 we prove the following complementary hardness results for functions f such that 3 f n n 3 the exact counting problem for f n regular bipartite graphs is p complete for and e 0 for any function f such that 3 f n n1 e approximate counting for f n regular bipartite graphs is as hard as approximate counting for all bipartite graphs an announcement of these results appears in dagum 1988

partitions irreducible characters and inequalities for generalized matrix functions given a partition a al a2 x xas x al a2 a of n we let xq denote the derived irreducible character of sn and we associate with a a derived partition a al a 2x x a tat l as 1 where t denotes the smallest positive integer such that at at i as 1 0 we show that if y is a decomposable c valued n linear function on cxm x cm x xcm n copies then x y y x y y translating into the notation of matrix theory we obtain an inequality involving the generalized matrix functions dx and dx namely that x e dxa b x e 1dx b for each n x n positive semidefinite hermitian matrix b this result generalizes a classical result of i schur and includes many other known inequalities as special cases

a fischer inequality for the second immanant abstract denote by h n the cone of n by n positive semidefinite hermitian matrices let d2 be the generalized matrix function or immanant afforded by the symmetric group sn and the irreducible degree n 1 character corresponding to the partition 2 1 1 suppose a h n is partitioned into blocks a a 11 a 12 a 12 a 22 if n 4 then d2 diag a11 a22 d2 a this follows from the fact that d2 is a schur concave function of the spectrum of a for a c n where c n consists of the matrices in h n whose diagonal entries all equal 1

immanantal invariants of graphs abstract something between an expository note and an extended research problem this article is an invitation to expand the existing literature on a family of graph invariants rooted in linear and multilinear algebra there are a variety of ways to assign a real n n matrix k g to each n vertex graph g so that g and h are isomorphic if and only if k g and k h are permutation similar it follows that g and h are isomorphic only if k g and k h are similar i e that similarity invariants of k g are graph theoretic invariants of g an observation that helps to explain the enormous literature on spectral graph theory the focus of this article is the permutation part i e on matrix functions that are preserved under permutation similarity if not under all similarity

computation of sparse circulant permanents via determinants we consider the problem of computing the permanent of circulant matrices we apply some recent results on the computation of the number of perfect matchings of small genus graphs in order to show that the permanent of a circulant matrix with three nonzero entries per row is the linear combination of just four determinants of circulant matrices with the same structure as the original matrix we also show that the same result holds true for a class of circulant matrices with four nonzero entries per row related to the dimer problem with periodic boundary conditions conversely we give hints at the fact that more general circulants do not share similar properties and thus should be dealt with by means of radically different approaches

continuous lattice effect algebras admitting order continuous states we prove that a necessary and sufficient condition for the existence of a faithful o continuous state on a complete modular atomic effect algebra e is the separability of e moreover we generalize the famous kaplansky theorem about order continuity of complemented complete modular lattices onto complete modular atomic effect algebras some other statements about the algebraic structure of modular effect algebras are shown we prove that every chain and every block in an irreducible complete modular atomic effect algebra is finite moreover every complete atomic modular effect algebra is compactly generated by finite elements

tensor product representations of general linear groups and their connections with brauer algebras abstract for the complex general linear group g gl r c we investigate the tensor product module t p v q v of p copies of its natural representation v c r and q copies of the dual spare v of v we describe the maximal vectors of t and from that obtain an explicit decomposition of t into its irreducible g summands knowledge of the maximal vectors allows us to determine the centralizer algebra c of all transformations on t commuting with the action of g to construct the irreducible c representations and to identify c with a certain subalgebra b r p q of the brauer algebra b r p q

hook immanantal inequalities for laplacians of trees abstract for an irreducible character x of the symmetric group sn indexed by the partition the immanant function d acting on an n n matrix a aij is defined as d a sn x ni 1 ai i the associated normalized immanant d is defined as d d x identity where identity is the identity permutation p heyfron has shown that for the partitions k 1n k the normalized immanant d k satisfies det a d 1 a d 2 a d n a per a 1 for all positive semidefinite hermitian matrices a when a is restricted to the laplacian matrices of graphs improvements on the inequalities above may be expected indeed in a recent survey paper r merris conjectured that d n 1 a n 2 n 1 d n a 2 wherever a is the laplacian matrix of a tree in this note we establish a refinement for the family of inequalities in 1 when a is the laplacian matrix of a tree that includes 2 as a special case these inequalities are sharp and equality holds if and only if a is the laplacian matrix of the star this is proved via the inequalities d k a d k 1 a d k 1 a d k a for k 2 3 n 1 where a is the laplacian matrix of a tree

how fast can one compute the permanent of circulant matrices abstract in this paper we address the problem of computing the permanent of 0 1 circulant matrices we investigate structural properties of circulant matrices showing that i if they are dense enough then they contain large arbitrary submatrices and ii if they are very sparse then they are not too far from convertible matrices building upon ii we then develop an efficient algorithm which allows us to compute permanents of very sparse circulants of size up to 200

on the orthogonal dimension of orbital sets let v be an inner product vector space over c and e1 e n an orthonormal basis of v a combinatorial necessary and sufficient condition for orthogonality of critical decomposable symmetrized tensors

inequalities for single hook immanants let dk denote the normalized generalized matrix function based upon the irreducible character of sn associated with the partition k 1 1 of n for all positive semidefinite hermitian n by n a we prove certain inequalities in the chain det a d1 a d2 a dn 1 a dn a per a conjectured for each n this work is in pursuit of the permanent on top conjecture and this conjecture is also verified for another class of immanants the methods involve several new inequalities for eigenvalues and the permanent which may be of independent interest

basic decomposition of elements and jauch piron effect algebras we show that every element of a complete atomic effect algebra e has a unique basic decomposition into a sum of a sharp element and unsharp multiples of isotropic atoms of e consequently for such effect algebras we obtain the smearing theorem for states establishing that every order continuous state existing on sharp elements of e can be extended to a state on e for a s complete separable atomic effect algebra e we prove that e is a unital and jauch piron effect algebra if and only if the set s e of all sharp elements of e is a unital jauch piron orthomodular lattice and for finite e s e is a boolean algebra

exterior products elementary symmetric functions and the fischer determinant inequality abstract if n and k are positive integers such that k h n the n n positive semidefinite hermitian matrices then det a det a k det a k for k 1 k e k a det a k det a k det a that is based on certain geometric invariants this analysis leads to various expansion formulas and a matrix theoretic refinement and extension of the fischer inequality in particular for each k such that 1 k h n then 0 k a det a k k a det a k det a k and i k a i 1 k a for each i such that 0 i k 1 as we demonstrate the matrix functions 0 k 1 k k k are closely related to the various laplace expansion formulas for the determinant function for example the inequality 0 k a 1 k a a h n translates into det a 1 k k i j 1 1 i j a ij det a i j where a aij and a i j denotes a with row i and column j deleted as a bonus we obtain inequalities for the higher order differences of the sequences 0 k a 1 k a k k a where a h n specifically we define the difference operator according to xi xi xi 1 and show that if a h n and q t k then 1 q q t k a 0

representations of the brauer algebra and littlewood s restriction rules let g be either sp v or o v using an action of the brauer algebra we describe the subspace t k v m v m of tensors of valence k as an induced representation as an application we recover a special case of littlewood s restriction rule affording the decomposition of an irreducible gl v module when restricted to g moreover we get an explicit realization of the irreducible representations of the brauer algebra m schur duality tells us that the algebra of operators generated by sp v and the above quotient of the brauer algebra are mutual central izer and establishes a bijective correspondence between the representations of either of these algebras the sp v module v m splits as v m m 2 k 0 t k v m the subspace t k v m being the sum of the sp v isotypic components of v m which appear for the first time in tensor power m 2k more directly if pq v m v m 2 is the extension operator which inserts in the positions p q the canonical element of the skew form t k v m is the vector space generated by k fold extensions of the traceless tensors in v m 2k i e

hook immanantal inequalities for trees explained abstract let d k denote the normalized hook immanant corresponding to the partition k ln k of n p heyfron proved the family of immanantal inequalities det a d 1 a d 2 a d n a per a 1 for all positive semidefinite hermitian matrices a motivated by a conjecture of r merris it was shown by the authors that 1 may be improved to d k 1 l t k 2 k 1 d k l t 2 for all 2 k n whenever l t is the laplacian matrix of a tree t the proof of 2 relied on rather involved recursive relations for weighted matchings in the tree t as well as identities of hook characters in this work we circumvent this tedium with a new proof using the notion of vertex orientations this approach makes 2 immediately apparent and more importantly provides an insight into why it holds namely the absence of certain vertex orientations for all trees as a by product we obtain an improved bound 0 1 k 1 d k l t d k l s n k 2 k 1 d k l t d k 1 l t where s n is the star with n vertices the ease with which the inequality in 2 and its improvement are derived points to the value of the concept of vertex orientation in the study of immanantal inequalities on graphs

the coefficients of the immanantal polynomial abstract an expression of the coefficient of immanantal polynomial of an n n matrix is present moreover we give expressions of the coefficient of immanantal polynomials of combinatorial matrices adjacency matrix laplacian matrix signless laplacian matrix as applications we show that the immanantal polynomials for laplacian matrix and signless laplacian matrix of bipartite graphs are the same this is a generalization of the characteristic polynomial for laplacian matrix and signless laplacian matrix of bipartite graphs furthermore we consider the relations between the characteristic polynomial and the immanantal polynomial for trees

characterization of homogeneity in orthocomplete atomic effect algebras we prove that homogeneity can be characterized by the decompositions of sharp elements in orthocomplete atomic effect algebras especially an orthocomplete atomic effect algebra is homogeneous if and only if the non zero coefficient of an atom in any atomic decomposition of all sharp elements is the same if and only if the non zero coefficient of an atom in any atomic decomposition of the unit element is the same which confirm the name homogeneous as an application we prove the state smearing theorem for orthocomplete atomic homogeneous effect algebras

the weyl approach to the representation theory of reflection equation algebra the present paper deals with the representation theory of reflection equation algebra connected to a hecke type r matrix up to some reasonable additional conditions the r matrix is arbitrary not necessary originating from quantum groups we suggest a universal method for constructing finite dimensional irreducible representations in the framework of the weyl approach well known in the representation theory of classical lie groups and algebras with this method a series of irreducible modules is constructed the modules are parametrized by young diagrams the spectrum of central elements sk trqlk is calculated in the single row and single column representations a rule for the decomposition of the tensor product of modules into a direct sum of irreducible components is also suggested

laplacian immanantal polynomials and the gts poset on trees abstract let t be a tree on n vertices with laplacian l t and let gts n be the generalised tree shift poset on the set of unlabelled trees on n vertices inequalities are known for coefficients of the characteristic polynomial of l t as we go up the poset gts n in this work we generalise these inequalities to the q laplacian l t q of t and to the coefficients of all immanantal polynomials

the number of n cycles in a graph abstract recently chang and fu y c chang h l fu the number of 6 cycles in a graph bull inst combin appl 39 2003 27 30 derived an exact expression based on powers of the adjacency matrix for the number of 6 cycles in a graph here i demonstrate a method for obtaining the number of n cycles in a graph from the immanants of the adjacency matrix the method is applicable to cycles of all sizes and to sets of disjoint cycles of any sizes and the cycles in the set need not be the same size

gradings on walled brauer algebras and khovanov s arc algebra abstract we introduce some z graded versions of the walled brauer algebra b r s working over a field of characteristic zero this allows us to prove that b r s is morita equivalent to an idempotent truncation of a certain infinite dimensional version of khovanov s arc algebra we deduce that the walled brauer algebra is koszul whenever 0

the holens dokovic conjecture on permanents fails abstract let a be a doubly stochastic matrix of order n and 1 a the sum of the order i subpermanents of a the holens okovic conjecture says in 1 a n i 1 2 i 1 a for each i 1 2 n there is a natural counterpart of this conjecture for 0 1 matrices with constant line sum k we show this associated conjecture holds when k 2 k n 2 i n k 1 or i 8 but that it fails in general because it wrongly asserts a polynomial bound on the ratio of near perfect to perfect matchings in k regular bipartite graphs

on majorization and schur products abstract suppose a d1 dm are n n matrices where a is self adjoint and let x m k 1 d k ad k it is shown that if d k d k d k d k i then the spectrum of x is majorized by the spectrum of a in general without assuming any condition on d1 dm a result is obtained in terms of weak majorization if each dk is a diagonal matrix then x is equal to the schur entrywise product of a with a positive semidefinite matrix thus the results are applicable to spectra of schur products of positive semidefinite matrices if a b are self adjoint with b positive semidefinite and if bii 1 for each i it follows that the spectrum of the schur product of a and b is majorized by that of a a stronger version of a conjecture due to marshall and olkin is also proved

weak multiplicativity for random quantum channels it is known that random quantum channels exhibit significant violations of multiplicativity of maximum output p norms for any p 1 in this work we show that a weaker variant of multiplicativity nevertheless holds for these channels for any constant p 1 given a random quantum channel mathcal n i e a channel whose stinespring representation corresponds to a random subspace s we show that with high probability the maximum output p norm of mathcal n otimes n decays exponentially with n the proof is based on relaxing the maximum output norm of mathcal n to the operator norm of the partial transpose of the projector onto s then calculating upper bounds on this quantity using ideas from random matrix theory

lattice effect algebras with o continuous faithful valuations we prove that if there exists an order continuous faithful valuation on a lattice effect algebra e then e is modular separable and order continuous it is also shown that such an effect algebra e can be supremum and infimum densely embedded into a complete effect algebra e which is also modular separable and order continuous since the valuation can be extended to a unique order continuous faithful valuation on e

hook immanantal and hadamard inequalities for q laplacians of trees abstract let t be a tree on n vertices with laplacian matrix l and q laplacian l q let k be the character of the irreducible representation of s n indexed by the hook partition k 1 n k and let d k l be the normalized hook immanant of l corresponding to the character k inequalities for d k l as k increases are known by assigning a statistic to vertex orientations on trees we generalize these inequalities to the q analogue l q of l for all q r and to the bivariate q t laplacian l q t for some values q t our statistic based approach also generalizes several other inequalities including the changing index k l of the hadamard inequality for l to the matrix l q and l q t thus we extend several results about l to l q which includes the case when l q is not positive semidefinite

minimum permanents on certain faces of matrices containing an identity submatrix abstract we determine the minimum permanents on certain faces of n for the fully indecomposable 0 1 matrices containing an identity submatrix of some order we also determine whether the given fully indecomposable 0 1 matrices are either cohesive and barycentric

almost all trees are co immanantal abstract if t is a tree denote by a t its adjacency matrix and by l t its laplacian matrix write d2 for the second immanant matrix function the main result is this let tn be the number of trees on n vertices let sn be the number of such trees t for which there is a nonisomorphic tree t such that d2 xi a t d2 xi a t and d2 xi l t d2 xi l t then sn tn 1 as n