Results 1  10
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15
Twenty combinatorial examples of asymptotics derived from multivariate generating functions
"... Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asym ..."
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Cited by 34 (14 self)
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Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morsetheoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have
Canonical characters on quasisymmetric functions and bivariate Catalan numbers, Electron
 J. Combin
"... Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character (Aguiar, Bergeron, and Sottile, math.CO/0310016). We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasisymmetric fun ..."
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Cited by 6 (0 self)
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Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character (Aguiar, Bergeron, and Sottile, math.CO/0310016). We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasisymmetric functions. They can be described in terms of Legendre’s beta function evaluated at halfintegers, or in terms of bivariate Catalan numbers: C(m, n) = (2m)!(2n)! m!(m + n)!n!. Properties of characters and of quasisymmetric functions are then used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients. Work supported in part by NSF grant DMS0302423 and by the NSF Postdoctoral Research Fellowship. We benefited from discussions with Ira Gessel and from the expertise of François Jongmans, who generously helped us search the 19th century literature in pursuit of a hardtofind article by Catalan. We also thank the referees for interesting remarks and suggestions. the electronic journal of combinatorics 11(2) (2005), #R15 1�
Algebraic evaluations of some Euler integrals, duplication formulae for Appell's hypergeometric function F_1, and Brownian variations
, 1999
"... Explicit evaluations of the symmetric Euler integral R 1 0 u ff (1 \Gamma u) ff f(u)du are obtained for some particular functions f . These evaluations are related to duplication formulae for Appell's hypergeometric function F1 which give reductions of F1(ff; fi; fi; 2ff; y; z) in terms of mor ..."
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Cited by 5 (3 self)
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Explicit evaluations of the symmetric Euler integral R 1 0 u ff (1 \Gamma u) ff f(u)du are obtained for some particular functions f . These evaluations are related to duplication formulae for Appell's hypergeometric function F1 which give reductions of F1(ff; fi; fi; 2ff; y; z) in terms of more elementary functions for arbitrary fi with z = y=(y \Gamma 1) and for fi = ff + 1 2 with arbitrary y; z. These duplication formulae generalize the evaluations of some symmetric Euler integrals implied by the following result: if a standard Brownian bridge is sampled at time 0, time 1, and at n independent random times with uniform distribution on [0; 1], then the broken line approximation to the bridge obtained from these n + 2 values has a total variation whose mean square is n(n + 1)=(2n + 1). Key words and phrases. Brownian bridge, Gauss's hypergeometric function, Lauricella's multiple hypergeometric series, uniform order statistics, Appell functions. AMS 1991 subject classification....
Range median of minima queries, super cartesian trees, and text indexing
"... A Range Minimum Query asks for the position of a minimal element between two specified arrayindices. We consider a natural extension of this, where our further constraint is that if the minimum in a query interval is not unique, then the query should return an approximation of the median position ..."
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Cited by 4 (3 self)
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A Range Minimum Query asks for the position of a minimal element between two specified arrayindices. We consider a natural extension of this, where our further constraint is that if the minimum in a query interval is not unique, then the query should return an approximation of the median position among all positions that attain this minimum. We present a succinct preprocessing scheme using only about 2.54 n + o(n) bits in addition to the static input array, such that subsequent “range median of minima queries” can be answered in constant time. This data structure can be constructed in linear time, and only o(n) additional bits are needed at construction time. We introduce several new combinatorial concepts such as SuperCartesian Trees and SuperBallot Numbers, which we believe will have other interesting applications in the future. We stress the importance of our result by giving two applications in text indexing; in particular, we show that our ideas are needed for fast construction of one component in Compressed Suffix Trees [19], a versatile tool for numerous tasks in text processing, and that they can be used for fast pattern matching in (compressed) suffix arrays [14].
A combinatorial interpretation for a superCatalan recurrence
 J. Integer Seq
"... Nicholas Pippenger and Kristin Schleich have recently given a combinatorial interpretation for the secondorder superCatalan numbers (un)n≥0 = (3, 2, 3, 6, 14, 36,...): they count “aligned cubic trees ” on n interior vertices. Here we give a combinatorial interpretation of the recurrence un = � n/ ..."
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Cited by 2 (1 self)
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Nicholas Pippenger and Kristin Schleich have recently given a combinatorial interpretation for the secondorder superCatalan numbers (un)n≥0 = (3, 2, 3, 6, 14, 36,...): they count “aligned cubic trees ” on n interior vertices. Here we give a combinatorial interpretation of the recurrence un = � n/2−1 k=0 �n−2 2k � 2 n−2−2k uk: it counts these trees by number of deep interior vertices where “deep interior ” means “neither a leaf nor adjacent to a leaf”. 1
A COMBINATORIAL INTERPRETATION OF THE NUMBERS 6(2n)!/n!(n + 2)!
, 2004
"... Abstract. It is well known that the numbers (2m)!(2n)!/m! n!(m+n)! are integers, but in general there is no known combinatorial interpretation for them. When m = 0 these numbers are the middle binomial coefficients () 2n, and when m = 1 they are twice the Catalan numbers. In this paper, we n give co ..."
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Cited by 1 (0 self)
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Abstract. It is well known that the numbers (2m)!(2n)!/m! n!(m+n)! are integers, but in general there is no known combinatorial interpretation for them. When m = 0 these numbers are the middle binomial coefficients () 2n, and when m = 1 they are twice the Catalan numbers. In this paper, we n give combinatorial interpretations for these numbers when m = 2 or 3.
Finding Range Minima in the Middle: Approximations and Applications
"... Abstract. A Range Minimum Query asks for the position of a minimal element between two specified arrayindices. We consider a natural extension of this, where our further constraint is that if the minimum in a query interval is not unique, then the query should return an approximation of the median ..."
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Abstract. A Range Minimum Query asks for the position of a minimal element between two specified arrayindices. We consider a natural extension of this, where our further constraint is that if the minimum in a query interval is not unique, then the query should return an approximation of the median position among all positions that attain this minimum. We present a succinct preprocessing scheme using Dn+o(n) bits in addition to the static input array (small constant D), such that subsequent “range median of minima queries” can be answered in constant time. This data structure can be built in linear time, with little extra space needed at construction time. We introduce several new combinatorial concepts such as SuperCartesian Trees and SuperBallot Numbers. We give applications of our preprocessing scheme in text indexes such as (compressed) suffix arrays and trees.
THE SUPER CATALAN NUMBERS S(m, m + s) FOR s ≤ 3 AND SOME INTEGER FACTORIAL RATIOS
"... Abstract. We give a combinatorial interpretation for the super Catalan number S(m, m + s) for s ≤ 3 using lattice paths and make an attempt at a combinatorial interpretation for s = 4. We also examine the integrality of some factorial ratios. ..."
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Abstract. We give a combinatorial interpretation for the super Catalan number S(m, m + s) for s ≤ 3 using lattice paths and make an attempt at a combinatorial interpretation for s = 4. We also examine the integrality of some factorial ratios.