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On the logconvexity of combinatorial sequences
 Advances in Applied Mathematics
"... Here presented is a survey for the logconvexity of some famous combinatorial sequences. We develop techniques for dealing with the logconvexity of sequences satisfying a threeterm recurrence. We also introduce the concept of qlogconvexity and establish the link with linear transformations prese ..."
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Here presented is a survey for the logconvexity of some famous combinatorial sequences. We develop techniques for dealing with the logconvexity of sequences satisfying a threeterm recurrence. We also introduce the concept of qlogconvexity and establish the link with linear transformations preserving the logconvexity. MSC: 05A20; 11B73; 11B83; 11B37
The Narayana Distribution
"... : The Narayana distribution is N n;k = 1 n i n k\Gamma1 ji n k j for 1 k n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining t ..."
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Cited by 21 (5 self)
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: The Narayana distribution is N n;k = 1 n i n k\Gamma1 ji n k j for 1 k n. This paper concerns structures counted by the Narayana distribution and bijective relationships between them. Here the statistic counting pairs of ascent steps on Catalan paths is prominently considered. Defining the n th Narayana polynomial as N n (w) = P 1kn N n;k w k , for n 1, the paper gives a combinatorial proof of a three term recurrence for these polynomials. It examines the Schroder numbers and the Kirkman numbers and establishes a sequence of bijections linking dissections of polygons to large Schroder paths. keywords: Catalan numbers, lattice paths, Schroder numbers AMS Subject Classification: 05A15 email: sulanke@math.idbsu.edu Note to typesetter: the symbol L is a calligraphic L. It should be set as either a calligraphic L or a script L. Thanks. 1 2 1. introduction Using the steps V = (0; 1) and H = (1; 0), the set of Catalan paths, Cat n , is the set of lattice paths from (0; ...
Generalizing Narayana and Schröder numbers to higher dimensions
 ELECTRON. J. COMBIN
, 2004
"... Let C(d, n) denotethesetofddimensional lattice paths using the steps X1:= (1, 0,...,0), X2: = (0, 1,...,0),...,Xd: = (0, 0,...,1), running from (0, 0,...,0) to (n,n,...,n), and lying in {(x1,x2,...,xd):0 ≤ x1 ≤ x2 ≤... ≤ xd}. Onanypath P: = p1p2...pdn ∈C(d, n), define the statistics asc(P):={i: pi ..."
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Cited by 14 (1 self)
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Let C(d, n) denotethesetofddimensional lattice paths using the steps X1:= (1, 0,...,0), X2: = (0, 1,...,0),...,Xd: = (0, 0,...,1), running from (0, 0,...,0) to (n,n,...,n), and lying in {(x1,x2,...,xd):0 ≤ x1 ≤ x2 ≤... ≤ xd}. Onanypath P: = p1p2...pdn ∈C(d, n), define the statistics asc(P):={i: pipi+1 = XjXℓ,j < ℓ}  and des(P):={i: pipi+1 = XjXℓ,j> ℓ}. Define the generalized Narayana number N(d, n, k) tocountthepathsinC(d, n) withasc(P)=k. We consider the derivation of a formula for N(d, n, k), implicit in MacMahon’s work. We examine other statistics for N(d, n, k) and show that the statistics asc and des −d +1 are equidistributed. We use Wegschaider’s algorithm, extending Sister Celine’s (WilfZeilberger) method to multiple summation, to obtain recurrences for N(3,n,k). We introduce the generalized large Schröder numbers (2d−1 k N(d, n, k)2k)n≥1 to count constrained paths using step sets which include diagonal steps.
Schröder Triangles, Paths, and Parallelogram Polyominoes.
, 1998
"... This paper considers combinatorial interpretations for two triangular recurrence arrays containing the Schroder numbers, s n = 1; 1; 3; 11; 45; ... and r n = 1; 2; 6; 22; 90; ... , for n = 0; 1; 2; .... These interpretations involve the enumeration of constrained lattice paths and bicolored parallel ..."
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Cited by 12 (2 self)
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This paper considers combinatorial interpretations for two triangular recurrence arrays containing the Schroder numbers, s n = 1; 1; 3; 11; 45; ... and r n = 1; 2; 6; 22; 90; ... , for n = 0; 1; 2; .... These interpretations involve the enumeration of constrained lattice paths and bicolored parallelogram polyominoes, called zebras. In addition to two recent inductive constructions of zebras and their associated generating trees, we present two new ones and a bijection between zebras and constrained lattice paths. We use the constructions with generating functions methods to count sets of zebras with respect to natural parameters.
Bijective Recurrences concerning Schröder paths
, 1998
"... Consider lattice paths in Z 2 with three step types: the up diagonal (1; 1), the down diagonal (1; \Gamma1), and the double horizontal (2; 0). For n 1, let S n denote the set of such paths running from (0; 0) to (2n; 0) and remaining strictly above the xaxis except initially and terminally. It ..."
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Consider lattice paths in Z 2 with three step types: the up diagonal (1; 1), the down diagonal (1; \Gamma1), and the double horizontal (2; 0). For n 1, let S n denote the set of such paths running from (0; 0) to (2n; 0) and remaining strictly above the xaxis except initially and terminally. It is well known that the cardinalities, r n = jS n j, are the large Schroder numbers. We use lattice paths to interpret bijectively the recurrence (n + 1)r n+1 = 3(2n \Gamma 1)r n \Gamma (n \Gamma 2)r n\Gamma1 , for n 2, with r 1 = 1 and r 2 = 2. We then use the bijective scheme to prove a result of Kreweras that the sum of the areas of the regions lying under the paths of S n and above the xaxis, denoted by AS n , satisfies AS n+1 = 6AS n \Gamma AS n\Gamma1 ; for n 2, with AS 1 = 1, and AS 2 = 7. Hence AS n = 1; 7; 41; 239; 1393; : : :. The bijective scheme yields analogous recurrences for elevated Catalan paths. Mathematical Reviews Subject Classification: 05A15 1 The paths and the recurr...
Bijective Recurrences for Motzkin Paths
, 2000
"... Consider lattice paths on Z² with steps (1; 1), (1; 1), and (1; 0). For n 2, let E n denote the set of such paths running from (0; 0) to (n; 0) and remaining strictly above the xaxis except initially and terminally. The cardinalities, f n = jE n j, are the Motzkin numbers, 1; 1; 2; 4; 9; 21; 51; ..."
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Cited by 7 (0 self)
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Consider lattice paths on Z² with steps (1; 1), (1; 1), and (1; 0). For n 2, let E n denote the set of such paths running from (0; 0) to (n; 0) and remaining strictly above the xaxis except initially and terminally. The cardinalities, f n = jE n j, are the Motzkin numbers, 1; 1; 2; 4; 9; 21; 51; 127; : : :, for n 2. We define a bijection yielding the recurrence (n + 1)f n+1 = (2n 1)f n + 3(n 2)f n 1 , for n 3. A modification of the bijection proves that the sum of the areas under the paths of E n , denoted by A n , satises A n+1 = 2A n +3A n 1 ; for n 3. A second modification yields a recurrence for a second moment on E n which agrees with Euler's recurrence for the central trinomial numbers.
Bijections for the Schröder Numbers
"... For n 1, each of the Schroder numbers, s n = 1; 1; 3; 11; 45; : : : counts the possible generalized bracketings on a word of n letters. Each of the large Schroder numbers, r n = 1; 2; 6; 22; 90; : : : , counts the lattice paths running from (0; 0) to (n \Gamma 1; n \Gamma 1), using the steps (0; 1) ..."
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For n 1, each of the Schroder numbers, s n = 1; 1; 3; 11; 45; : : : counts the possible generalized bracketings on a word of n letters. Each of the large Schroder numbers, r n = 1; 2; 6; 22; 90; : : : , counts the lattice paths running from (0; 0) to (n \Gamma 1; n \Gamma 1), using the steps (0; 1), (1; 1), and (1; 0), and never passing below the line y = x. We give a bijective explanation that relates these structures and shows 2s n = r n , for n 2.
Positivity of threeterm recurrence sequences
, 2010
"... In this paper, we give the sufficient conditions for the positivity of recurrence sequences defined by anun = bnun−1 − cnun−2 for n � 2, where an,bn,cn are all nonnegative and linear in n. As applications, we show the positivity of many famous combinatorial sequences. ..."
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In this paper, we give the sufficient conditions for the positivity of recurrence sequences defined by anun = bnun−1 − cnun−2 for n � 2, where an,bn,cn are all nonnegative and linear in n. As applications, we show the positivity of many famous combinatorial sequences.
Parametric Catalan numbers and Catalan triangles
 Linear Algebra Appl
"... Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Belltype Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan n ..."
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Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Belltype Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schröder numbers and small Schröder numbers, respectively. Using the characterization sequences of Belltype Riordan arrays, some properties and expressions including the Taylor expansions of generalized Catalan numbers are given. A few combinatorial interpretations of the generalized Catalan numbers are also provided. Finally, a generalized Motzkin numbers and Motzkin triangles are defined similarly. An interrelationship among parametrical Catalan triangle, Pascal triangle, and