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In this section we consider a surprising application of certain adjacency ma_trices to some problems in extremal set theory. An important role will also be played by �nite groups. In general, extremal set theory is concerned with �nding (or estimating) the most or least number of sets satisfying given set-theoretic or combinatorial conditions.

This paper is primarily intended as an introduction for the mathematically inclined to some of the rich algebraic combinatorics arising in for instance CFT. It is essentially self-contained, apart from some of the background motivation and examples which are included to give the reader a sense of the context. The theory is still a work-in-progress, and emphasis is given here to several open questions and problems.

Journal of Algebraic Combinatorics 1993: Volume 2 , Issue CONTENTS. Digitized from IA1574829-03 . Previous issue: sim_journal-of-algebraic-combinatorics_1992-12_1_4 . Next issue: sim_journal-of-algebraic-combinatorics_1993-03_2_1 .

Journal of Algebraic Combinatorics 1994: Volume 3 , Issue CONTENTS. Digitized from IA1574829-03 . Previous issue: sim_journal-of-algebraic-combinatorics_1993-11_2_4 . Next issue: sim_journal-of-algebraic-combinatorics_1994-01_3_1 .

Journal of Algebraic Combinatorics 1994: Volume 3 , Issue CONTENTS. Digitized from IA1574829-03 . Previous issue: sim_journal-of-algebraic-combinatorics_1993-11_2_4 . Next issue: sim_journal-of-algebraic-combinatorics_1994-01_3_1 .

Journal of Algebraic Combinatorics is an active publication covering algebraic combinatorics. It was established in 1992 and is published 8 times a year by Springer New York LLC. The journal serves as a resource for researchers in the field of combinatorics, as well as mathematical and computer scientists with an interest in combinatorial structure. It publishes articles that cover a wide range of topics, including but not limited to: graph theory, matroid theory, combinatorial optimization, representation theory, and algebraic geometry. The journal welcomes both theoretical and applied contributions./p> Journal of Algebraic Combinatorics can be found online at: https://www.springer.com/journal/10801

We show, in great detail, how the perturbative tools of quantum field theory allow one to rigorously obtain: a ``categorified'' Faa di Bruno type formula for multiple composition, an explicit formula for reversion and a proof of Lagrange-Good inversion, all in the setting of multivariable power series. We took great pains to offer a self-contained presentation that, we hope, will provide any mathematician who wishes, an easy access to the wonderland of quantum field theory.

De�nition: If a vertex v has at least as many chips on it as its degree, i.e. C(v) � deg(v), we say that v is ready to �re. A vertex v that is ready to �re may send chips to its neighbors by sending one chip along each of its incident edges. Given graph G, we say that a chip con�guration C is stable if no vertex is ready to �re, i.e. if C(v) < deg(v) for all vertices v _ V (G).

Theorem 1: Let G be a directed acyclic graph with n designated origin and destination nodes, and let A be the n-by-n matrix whose (i, j)-entry is the number of paths from the ith origin to the jth destination. The following statements hold: (a) The number of n-paths is equal to the permanent of A. (b) If G is nonpermutable, the number of nonintersecting n-paths is equal to the determinant of A. (c) In general, even if G is not nonpermutable, the determinant of A equals Even(G) _ Odd(G), where Even(G) is the number of nonintersecting n-paths corresponding to even permutations and Odd(G) is the number of nonintersecting n-paths corresponding to odd permutations.

A partition of n is an ordered set of positive integers _ = (_1, _2, . . . , _n) such that _i = n and _1 � _2 � � � � � _

A partition of n is an ordered set of positive integers _ = (_1, _2, . . . , _n) such that _i = n and _1 � _2 � � � � � _

A partition of n is an ordered set of positive integers _ = (_1, _2, . . . , _n) such that _i = n and _1 � _2 � � � � � _

A partition of n is an ordered set of positive integers _ = (_1, _2, . . . , _n) such that _i = n and _1 � _2 � � � � � _

The following material can be found in a number of sources, including Sections 7.3 _ 7.5, 7.7, 7.10 _ 7.11, 7.15 _ 16 of Stanley�s Enumerative Combinatorics Volume 2. Elementary and Homogeneous Symmetric Func_tions: A polynomial in n variables, P (x1, x2, . . . , xn) _ C[x1, x2, . . . , xn] is known as a symmetric polynomial if for any permutation _ _ Sn, P(x_(1), x_(2), . . . , x_(n)) = P(x1, x2, . . . , xn).

De�nition: An anitchain A of a poset P is a subset of elements of P such that for all x, y _ A, x _� y and y _� x. We denote the levels of a graded poset P as Pi where Pi = {x _ P : rank(x) = i}.

De�niton: A Perfect Matching of a graph G = (V, E) is a set M _ E of distringuished edges such that each vertex v _ V is incident to exactly one edge of M.

We describe how to construct and enumerate Magic squares, Franklin squares, Magic cubes, and Magic graphs as lattice points inside polyhedral cones using techniques from Algebraic Combinatorics. The main tools of our methods are the Hilbert Poincare series to enumerate lattice points and the Hilbert bases to generate lattice points. We define polytopes of magic labelings of graphs and digraphs, and give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.

We describe how to construct and enumerate Magic squares, Franklin squares, Magic cubes, and Magic graphs as lattice points inside polyhedral cones using techniques from Algebraic Combinatorics. The main tools of our methods are the Hilbert Poincare series to enumerate lattice points and the Hilbert bases to generate lattice points. We define polytopes of magic labelings of graphs and digraphs, and give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.

Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type B noncommutative symmetric functions, noncommutative supersymmetric functions and colored free quasi-symmetric functions.

Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type B noncommutative symmetric functions, noncommutative supersymmetric functions and colored free quasi-symmetric functions.

This chapter introduces and elaborates on the fruitful interplay of coding theory and algebraic combinatorics, with most of the focus on the interaction of codes with combinatorial designs, finite geometries, simple groups, sphere packings, kissing numbers, lattices, and association schemes. In particular, special interest is devoted to the relationship between codes and combinatorial designs. We describe and recapitulate important results in the development of the state of the art. In addition, we give illustrative examples and constructions, and highlight recent advances. Finally, we provide a collection of significant open problems and challenges concerning future research.

In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.

This material closely follows selections from Chapter 3 of Enumerative Combinatorics 1 by Richard Stanley

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield--Polya theory, Witt vectors, and totally nonnegative matrices; resultants; discriminants (up to quartics); Macdonald polynomials; key polynomials; Demazure atoms; Schubert polynomials; and Grothendieck polynomials, among others. Our principal construction is the Schubitope. For any subset of [n] x [n], we describe it by linear inequalities. This generalized permutahedron conjecturally has positive Ehrhart polynomial. We conjecture it describes the Newton polytope of Schubert and key polynomials. We also define dominance order on permutations and study its poset-theoretic properties.

Theorem (Schensted): Let � be a permutation of {1, 2, . . . , n} written in one-line notation. Let P and Q be the Standard Young Tableaux (SYT) in the image of the (Robinson-Schensted-Knuth) RSK algorithm, i.e. RSK(�) = (P, Q), with shapes sh(P) = sh(Q) = _. Then the length of the longest increasing subsequence in � equals the length of the �rst row of _ and the length of the longest decreasing subsequence in � equals the length of the �rst column of _.

Today�s lecture notes cover the Oriented Matrix Theorem, which is discussed in Sections 9 and 10 of Richard Stanley�s "Topics in Algebraic Combinatorics� lecture notes. The proof presented in class more closely resembles the bijective proof in Section 4.4 of "Constructive Combinatorics� by Dennis Stanton and Dennis White. I also thank Adriano Garsia for introducing me to this proof.

Recurrence Relations and Generating Functions: Given an in�nite sequence of numbers, a generating function is a compact way of expressing this data. We begin with the notion of ordinary generating functions.

Recurrence Relations and Generating Functions: Given an in�nite sequence of numbers, a generating function is a compact way of expressing this data. We begin with the notion of ordinary generating functions.

The following is an outline of the material covered April 6th and 8th in class. This material can be found in Chapter 5 of Stanley�s Enumerative Combinatorics Volume 2. Proofs of most of the results are in class notes.

The following is an outline of the material covered April 6th and 8th in class. This material can be found in Chapter 5 of Stanley�s Enumerative Combinatorics Volume 2. Proofs of most of the results are in class notes.

Journal of Algebraic Combinatorics 1992: Volume 1 , Issue CONTENTS. Digitized from IA1574829-03 . Next issue: sim_journal-of-algebraic-combinatorics_1992-05_1_1 .

We obtain a necessary and sufficient condition for the linear independence of solutions of differential equations for hyperlogarithms. The key fact is that the multiplier (i.e. the factor $M$ in the differential equation $dS=MS$) has only singularities of first order (Fuchsian-type equations) and this implies that they freely span a space which contains no primitive. We give direct applications where we extend the property of linear independence to the largest known ring of coefficients.

A survey written for the upcoming "Handbook of Enumerative Combinatorics".

Journal of Algebraic Combinatorics 1996: Volume 5 , Issue CONTENTS. Digitized from IA1574829-03 . Previous issue: sim_journal-of-algebraic-combinatorics_1995-10_4_4 . Next issue: sim_journal-of-algebraic-combinatorics_1996-01_5_1 .

Journal of Algebraic Combinatorics 1996: Volume 5 , Issue CONTENTS. Digitized from IA1574829-03 . Previous issue: sim_journal-of-algebraic-combinatorics_1995-10_4_4 . Next issue: sim_journal-of-algebraic-combinatorics_1996-01_5_1 .

Journal of Algebraic Combinatorics 1996: Volume 5 , Issue CONTENTS. Digitized from IA1574829-03 . Previous issue: sim_journal-of-algebraic-combinatorics_1995-10_4_4 . Next issue: sim_journal-of-algebraic-combinatorics_1996-01_5_1 .

We consider generators of algebraic covariant derivative curvature tensors R' which can be constructed by a Young symmetrization of product tensors W*U or U*W, where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2,1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if W is symmetric or skew-symmetric, respectively. In special cases these numbers can be reduced to the minima 12 or 10. If these minima occur then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of symmetric or alternating 2-tensor fields. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[S_r] and discrete Fourier transforms for symmetric groups S_r. For symbolic calculations we used the Mathematica packages Ricci and PERMS.

Journal of Algebraic Combinatorics 1995: Volume 4 , Issue CONTENTS. Digitized from IA1574829-03 . Previous issue: sim_journal-of-algebraic-combinatorics_1994-10_3_4 . Next issue: sim_journal-of-algebraic-combinatorics_1995-01_4_1 .

Properties of a given symmetry group G are very important in investigation of a physical system invariant under its action. In the case of finite spin systems (magnetic rings, some planar macromolecules) the symmetry group is isomorphic with the dihedral group D_N. In this paper group-theoretical `parameters' of such groups are determined, especially decompositions of transitive representations into irreducible ones and double cosets. These results are necessary to construct matrix elements of any operator commuting with G in an efficient way. The approach proposed can be usefull in many branches of physics, but here it is applied to finite spin systems, which serve as models for mesoscopic magnets.

A survey of recent progress in three areas of algebraic combinatorics: (1) the Saturation Conjecture for Littlewood-Richardson coefficients, (2) the n! and (n+1)^{n-1} conjectures, and (3) longest increasing subsequences of permutations.

A survey of recent progress in three areas of algebraic combinatorics: (1) the Saturation Conjecture for Littlewood-Richardson coefficients, (2) the n! and (n+1)^{n-1} conjectures, and (3) longest increasing subsequences of permutations.

We give a brief account of a construction called tokens here, which is significant in algebra, analysis, combinatorics, and physics. Tokens allow to express a semigroup on one set via a semigroup convolution on another set. Therefore tokens are similar to intertwining operators but are more flexible. Keywords: semigroups, hypergroups, tokens, poset, multiplicative functions, polynomial sequence of binomial type, integral kernel, wavelets, refinement equation, special functions, quantum propagator, path integral, quantum computing.

We give a brief account of a construction called tokens here, which is significant in algebra, analysis, combinatorics, and physics. Tokens allow to express a semigroup on one set via a semigroup convolution on another set. Therefore tokens are similar to intertwining operators but are more flexible. Keywords: semigroups, hypergroups, tokens, poset, multiplicative functions, polynomial sequence of binomial type, integral kernel, wavelets, refinement equation, special functions, quantum propagator, path integral, quantum computing.

The stabilization algorithm of Weisfeiler and Leman has as an input any square matrix A of order n and returns the minimal cellular (coherent) algebra W(A) which includes A. In case when A=A(G) is the adjacency matrix of a graph G the algorithm examines all configurations in G having three vertices and, according to this information, partitions vertices and ordered pairs of vertices into equivalence classes. The resulting construction allows to associate to each graph G a matrix algebra W(G):= W(A(G))$ which is an invariant of the graph G. For many classes of graphs, in particular for most of the molecular graphs, the algebra W(G) coincides with the centralizer algebra of the automorphism group aut(G). In such a case the partition returned by the stabilization algorithm is equal to the partition into orbits of aut(G). We give algebraic and combinatorial descriptions of the Weisfeiler--Leman algorithm and present an efficient computer implementation of the algorithm written in C. The results obtained by testing the program on a considerable number of examples of graphs, in particular on some chemical molecular graphs, are also included.

The spin correlations \omega^z_r, r=1,2,3, and the probability p_N$ of finding a system in the Neel state for the antiferromagnetic ring Fe(III)6 (the so-called `small ferric wheel') are calculated. States with magnetization M=0, total spin 0