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35
Conjectures on the quotient ring by diagonal invariants
 J. ALGEBRAIC COMBIN
, 1994
"... We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring Q[x1,...,xn,y1,...,yn] in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x1,.. ..."
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Cited by 96 (10 self)
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We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring Q[x1,...,xn,y1,...,yn] in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = {x1,...,xn} is classical. Introducing the second set of variables leads to a ring about which little is yet understood, but for which there is strong evidence of deep connections with many fundamental results of enumerative combinatorics, as well as with algebraic geometry and Lie theory.
A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron
"... The volume of the ndimensional polytope for arbitrary x := (x 1 ; : : : ; x n ) with x i > 0 for all i de nes a polynomial in variables x i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this po ..."
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Cited by 40 (2 self)
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The volume of the ndimensional polytope for arbitrary x := (x 1 ; : : : ; x n ) with x i > 0 for all i de nes a polynomial in variables x i which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two dierent polytopal subdivisions of n (x). The rst of these subdivisions generalizes to a class of polytopes called sections of order cones. In the second subdivision, the chambers are indexed in a natural way by rooted binary trees with n + 1 vertices, and the con guration of these chambers provides a representation of another polytope with many applications, the associahedron.
Parking Functions and Noncrossing Partitions
 Electronic J. Combinatorics
, 1997
"... this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions ..."
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Cited by 29 (5 self)
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this paper we will develop a connection between parking functions and another topic, viz., noncrossing partitions
A Remarkable q,tCatalan Sequence and qLagrange Inversion
"... We introduce a rational function C n (q; t) and conjecture that it always evaluates to a polynomial in q; t with nonnegative integer coefficients summing to the familiar Catalan number \Gamma 2n \Delta . We give supporting evidence by computing the specializations D n (q) = C n (q; 1=q) ..."
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Cited by 21 (7 self)
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We introduce a rational function C n (q; t) and conjecture that it always evaluates to a polynomial in q; t with nonnegative integer coefficients summing to the familiar Catalan number \Gamma 2n \Delta . We give supporting evidence by computing the specializations D n (q) = C n (q; 1=q) q and C n (q) = C n (q; 1) = C n (1; q). We show that, in fact, D n (q) qcounts Dyck words by the major index and C n (q) qcounts Dyck paths by area. We also show that C n (q; t) is the coefficient of the elementary symmetric function e n in a symmetric polynomial DH n (x; q; t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that C n (q; t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DH n (x; q; 1) yields a novel and combinatorial way of expressing the solution of the qLagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis fP (x; q; t)g which are best dealt with inring notation. In particular we derive here thering version of several symmetric function identities.
An Introduction to Hyperplane Arrangements
 Lecture notes, IAS/Park City Mathematics Institute
, 2004
"... ..."
Parking Functions, Empirical Processes, and the Width of Rooted Labeled Trees
"... This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking f ..."
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Cited by 20 (5 self)
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This paper provides tight bounds for the moments of the width of rooted labeled trees with n nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many onetoone correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
Free quasisymmetric functions of arbitrary level
"... Abstract. We introduce analogues of the Hopf algebra of Free quasisymmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products Γ ≀ Sn and the corresponding generalizations of quasisymmetric functio ..."
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Cited by 18 (9 self)
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Abstract. We introduce analogues of the Hopf algebra of Free quasisymmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products Γ ≀ Sn and the corresponding generalizations of quasisymmetric functions. Also, we obtain Hopf algebras of colored parking functions. 1.
Linear Probing and Graphs
 Algorithmica
, 1997
"... . Mallows and Riordan showed in 1968 that labeled trees with a small number of inversions are related to labeled graphs that are connected and sparse. Wright enumerated sparse connected graphs in 1977, and Kreweras related the inversions of trees to the socalled "parking problem" in 1980. A combina ..."
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Cited by 17 (0 self)
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. Mallows and Riordan showed in 1968 that labeled trees with a small number of inversions are related to labeled graphs that are connected and sparse. Wright enumerated sparse connected graphs in 1977, and Kreweras related the inversions of trees to the socalled "parking problem" in 1980. A combination of these three results leads to a surprisingly simple analysis of the behavior of hashing by linear probing, including higher moments of the cost of successful search. The wellknown algorithm of linear probing for n items in m ? n cells can be described as follows: Begin with all cells (0; 1; : : : ; m \Gamma 1) empty; then for 1 k n, insert the kth item into the first nonempty cell in the sequence h k ; (h k + 1) mod m; (h k + 2) mod m; : : : , where h k is a random integer in the range 0 h k ! m. (See, for example, [4, Algorithm 6.4L].) The purpose of this note is to exhibit a surprisingly simple solution to a problem that appears in a recent book by Sedgewick and Flajolet [9]: E...