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Super ballot numbers
Abstract
The Catalan numbers are are well-known integers that arise in many combinatorial problems. The numbers , , and more generally are also integers for all n. We study the properties of these numbers and of some analogous numbers that generalize the ballot numbers, which we call super ballot numbers.
... by equation (30) of [6]. Additionally, when r = u and l = d, Formula (1) becomes ...
... which is known [6,Equation 29] to be the super Catalan number, S(r, k) = (2r)!(2k)! r!k!(r + k)! . ...
... Note that 2Bin j (2j − 1) = Bin j (2j) for j ∈ N.6 We do not require the coefficient of each Bin j (2j − 1) to be even, as is the case when a = b and c = d. In fact, such a requirement would make Conjecture 4.4 untrue. ...
We resolve a conjecture of Albert and Bousquet-Melou enumerating quarter-plane walks with fixed horizontal and vertical projections according to their upper-right-corner count modulo 2. In doing this, we introduce a signed upper-right-corner count statistic. We find its distribution over planar walks with any choice of fixed horizontal and vertical projections. Additionally, we prove that the polynomial counting loops with a fixed horizontal and vertical projection according to the absolute value of their signed upper-right-corner count is (x+1)-positive. Finally, we conjecture an equivalence between (x+1)-positivity of the generating function for upper-right-corner count and signed upper-right-corner count.
... Interestingly, these numbers are not necessarily integers but the numbers given by 6 (2n)! n!(n + 2)! do form an integer sequence. In 1992, Gessel [7] showed that, in fact, the generalized Catalan numbers J r (2n)! n!(n + r + 1)! are integers when J r is chosen to be (2r + 1)!/r!. ...
... m!n!(m + n)! are integers, but there is no known combinatorial interpretation for them in general. Gessel [7] called these numbers the super Catalan numbers since S(1, n)/2 gives the Catalan number C n . Note that S(2, n)/2 = 6 (2n)! n!(n+2)! . ...
... problem, yet will likely prove challenging given that there is a combinatorial interpretation for the super Catalan numbers in only a handful of cases. In addition, the super Catalan numbers satisfy a number of interesting binomial identities, such as this identity of von Szily (1894), which can be found in [7], Eq. (29), p. 11: S(m, n) = k∈Z (−1) k 2m m + k 2n n + k . ...
Catalan observed in 1874 that the numbers , now called the super Catalan numbers, are integers but there is still no known combinatorial interpretation for them in general, although interpretations have been given for the case m=2 and for for . In this paper, we define the super FiboCatalan numbers and prove they are integers for m=1 and m=2. In addition, we prove that is an integer for .
... For m = 1, all four sums (1), (2), (3), and (4) reduce to interesting combinatorial identities. For example, it is well-known that [ Recently, the following two binomial coefficient identities involving the Catalan numbers were discovered [8]: ...
... First, note that all four sums (1), (2), (3), and (4) are instances of S(n, m). For m ≥ 2, by Eq. (8), it follows that S(n, m) = D S (n, 0, m − 2). ...
... In this paper, we show how the method of D sums works on harder sums, such as our four sums (1), (2) Let S, F , and D S be sums according to Definition 6. The main obstacle is to calculate the sum D S (n, j, 0). ...
We prove that some sums, which arise as generalizations of known binomial coefficient identities, are divisible by the central binomial coefficient. A new method is used. In particular, we show that an alternating sum concerning the product of a power of a binomial coefficient with two Catalan numbers is always divisible by the central binomial coefficient.
... For m = 1, all four sums (1), (2), (3), and (4) reduce to interesting combinatorial identities. For example, it is well-known that [ Recently, the following two binomial coefficient identities involving the Catalan numbers were discovered [8]: ...
... First, note that all four sums (1), (2), (3), and (4) are instances of S(n, m). For m ≥ 2, by Eq. (8), it follows that S(n, m) = D S (n, 0, m − 2). ...
... In this paper, we show how the method of D sums works on harder sums, such as our four sums (1), (2) Let S, F , and D S be sums according to Definition 6. The main obstacle is to calculate the sum D S (n, j, 0). ...
... Some elementary combinatorial properties of the Catalan and ballot numbers were given in [2,6,7]. The ...
... Catalan numbers are used in many mathematical problems [2,7]. ...
... In [2], the Catalan numbers are special cases of the ballot numbers ...
... In [3], Gessel shows that T (m, n) is a positive integer for all (m, n) ∈ Z ≥0 × Z >0 . It is natural to ask if a combinatorial interpretation of T (m, n) exists. ...
... (3) T (3, n) = G n (3, 1) + 2G n−1 (3,0) Theorem 1 is used to prove: ...
... Since G n (m, k) is defined combinatorially in (2), (3) and (4) immediately imply a combinatorial interpretation for T (3, n) and T (4, n). ...
The Super-Catalan numbers are a generalization of the Catalan numbers defined as . It is an open problem to find a combinatorial interpretation for T(m,n). We resolve this for m=3,4 using a common form; no such solution exists for m=5.
... Here (A127779, A104633) are in reciprocal pair. In particular, A033820 is connected to the enumeration of paths avoiding the line x = y; see [114,218]. ...
... which proves the first recurrence in (114). On the other hand, when n is odd, we construct n 2 ! ...
... ways to permutes these blocks), and then append an element 2σ n 2 − 1 to the tail, yielding parity alternating permutations with the required property. Since this construction is reversible, this proves (114) in the odd case. ...
We study linear recurrences of Eulerian type of the formPn(v)=(α(v)n+γ(v))Pn−1(v)+β(v)(1−v)Pn−1′(v)(n⩾1), with P0(v) given, where α(v),β(v) and γ(v) are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of Pn(v) for large n using the method of moments and analytic combinatorial tools under varying α(v),β(v) and γ(v), and apply our results to more than two hundred of concrete examples when β(v)≠0 and more than three hundred when β(v)=0 that we gathered from the literature and from Sloane's OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.
... In this paper we study numbers P (m, n) which satisfy the summation equations 1 The switching of m and n on the right hand side is due to other equalities explained later on. 1 which exactly have the solutions B(m, n) = (−1) n (c 1 n + c 0 ) with c 1 , c 0 ∈ C for each single m. Notice further that A(n, m) for fixed m is polynomial in n of degree 2(m − 1). ...
... In this paper we study numbers P (m, n) which satisfy the summation equations 1 The switching of m and n on the right hand side is due to other equalities explained later on. 1 which exactly have the solutions B(m, n) = (−1) n (c 1 n + c 0 ) with c 1 , c 0 ∈ C for each single m. Notice further that A(n, m) for fixed m is polynomial in n of degree 2(m − 1). ...
... By virtue of formula (1), the values P (m, n) = P (n, m) + (−1) m (c 1 (n)m + c 0 (n)) give a solution of (2) for all c 0 (n), c 1 (n) ∈ C. In Theorem 3.1 we show that indeed P (m, n) also is a solution of (2). It follows that P (m, n) is nearly symmetric for all m, n ∈ N, P (m, n) = P (n, m) + (−1) m c 1 (n)m + c 0 (n) , and we show c 1 (n) = (−1) n−1 and c 0 (n) = (−1) n · n. ...
We give several families of polynomials which are related by Eulerian summation operators. They satisfy interesting combinatorial properties like being integer-valued at integral points. This involves nearby-symmetries and a recursion for the values at half-integral points. We also obtain identities for super Catalan numbers.
... Note that S(1, n)/2 = C n is the usual Catalan number. The problem of finding a combinatorial interpretation was posed by Gessel in [72]. Such an interpretation is known for m ≤ 3 and for |m − n| ≤ 3, see [35,74] (see also [8,152]). ...
... Proof. Following Gessel [72], we have ...
... 66]. Gessel writes: "it remains to be seen whether (4.1) can be interpreted in a 'natural' way" [72]. This was later echoed in [74]: "Formula (4.1) allows us to construct recursively a set of cardinality S(m, n), but it is difficult to give a natural description of this set." ...
We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.
... There are several binomial coefficient identities for super Catalan numbers. For example, the identity of von Szily (1894): [7,Eq. (29), p. 11] S(n, l) = k∈Z (−1) k 2n n + k 2l l + k . ...
... Note that the identity of von Szily gives another proof that the number S(n, l) is always an integer. See also [7,Eq. (31); Eq. (32), p. 12]. ...
We present a new alternating convolution formula for the super Catalan numbers which arises as a generalization of two known binomial identities. We prove a generalization of this formula by using auxiliary sums, recurrence relations, and induction. By using a new method, we prove one interesting divisibility result with super Catalan numbers.
... There are several binomial coefficient identities for super Catalan numbers. For example, the identity of von Szily (1894): [7,Eq. (29), p. 11] S(n, l) = k∈Z (−1) k 2n n + k 2l l + k . ...
... Note that the identity of von Szily gives another proof that the number S(n, l) is always an integer. See also [7,Eq. (31); Eq. (32), p. 12]. ...
We present a new alternating convolution formula for the super Catalan numbers which arises as a generalization of two known binomial identities. We prove a generalization of this formula by using auxiliary sums, recurrence relations, and induction. By using a new method, we prove one interesting divisibility result with super Catalan numbers.
... Before we get to the main discussion of this paper, we want to mention some recent developments in the research on the super Catalan numbers. These numbers were perhaps first brought back to the public's attention by Gessel in his paper [10]. First we note from the definition of the super Catalan numbers in (1) that S (m, n) is always even for every m and n except when m = n = 0, so sometimes the numbers T (m, n) ≡ 1 2 S(m, n) are studied instead. ...
... so it is sufficient to prove the uniqueness of I (m, 0) = φ b (α 2m ). By using equation (10) in Theorem 8 and the double-parity property, we can again make a change of index s → 2s to get ...
We study polynomial summation over unit circles over finite fields of odd characteristic, obtaining a purely algebraic integration theory without recourse to infinite procedures. There are nonetheless strong parallels to classical integration theory over a circle, and we show that the super Catalan numbers and closely related rational numbers lie at the heart of both theories. This gives a uniform analytic meaning to these up to now somewhat mysterious numbers.
Our derivation utilises the three-fold symmetry of chromogeometry between Euclidean and relativistic geometries, and we find that the Fourier summation formulas we derive in these two different settings are closely connected.
... The some elementary combinatorial properties of the Catalan and Ballot numbers are given in [2], [4] and [3]. In [1], [7],É. ...
... The Catalan numbers are given by In [2], the Catalan numbers are special cases of the Ballot numbers B .n; k/ D k 2n C k 2n C k n ! : ...
... Notice that the cardinality of D ′ n is S(n, 2)/2 where S(n, m) = (2n)!(2m)! n!m!(n+m)! is the nth super-Catalan number of order m (these numbers are discussed by Gessel in [67]). Our bijection gives an interpretation of these numbers for m = 2. ...
... Similary, given a univariate generating function f (x) = n f n x n , write [x n ]f (x) for the coefficient f n . The bivariate version of Lagrange inversion formula [67] states that, if two bivariate generating functions B(x, y) and W (x, y) are related by a system of the form B(x, y) = xφ(W (x, y)) W (x, y) = yψ(B(x, y)), n−1 . The coefficient B n,r,k can be derived in a similar way. ...
This thesis describes algorithms on planar maps (graphs embedded in the plane without edge-crossings) based on their combinatorial properties. For several important families of planar maps (3-connected, triangulations, quadran- gulations), efficient procedures of random generation, encoding, and straight-line drawing are described. In particular, the first optimal encoder for the combinatorial incidences of polygonal meshes with spherical topology is developed. Starting from a generator for 3-connected maps, a new random generator for planar graphs is in- troduced. The complexity of generation is the best currently known: quadratic (in expectation) for exact-size sampling and linear (in expectation) for approximate- size sampling. Finally, several straight-line drawing algorithms for planar maps are introduced. The procedures are both simple to describe and very efficient, yielding the best known grid size for two families of maps: triangulations of the 4-gon with no filled 3-cycle —called irreducible— and quadrangulations. The algo- rithms presented in the thesis take advantage of several combinatorial structures on planar maps (orientations, partitions into spanning trees) as well as new bijective constructions
... (4) Gessel's [7] super ballot numbers (often also called super-Catalan numbers) (number of rooted Hamiltonian maps with 2n vertices; cf. [19]), to mention just a few. ...
We prove a central limit theorem for the joint distribution of , , where denotes the sum-of-digits function in base~q and the 's are positive integers relatively prime to q. We do this in fact within the framework of quasi-additive functions. As application, we show that most elements of "Catalan-like" sequences - by which we mean integer sequences defined by products/quotients of factorials - are divisible by any given positive integer.
... An alternate approach to the generating function is to use formula 2.3 to obtain what MacMahon called a "redundant generating function " (cf. [10]), since it contains terms other than those which are combinatorially significant. ...
Catalan numbers enumerate binary trees and Dyck paths. The distribution of paths with respect to their number k of factors is given by ballot numbers . These integers are known to satisfy simple recurrence, which may be visualised in a ``Catalan triangle'', a lower-triangular two-dimensional array. It is surprising that the extension of this construction to 3 dimensions generates integers that give a 2-parameter distribution of , which may be called order-3 Fuss-Catalan numbers, and enumerate ternary trees. The aim of this paper is a study of these integers . We obtain an explicit formula and a description in terms of trees and paths. Finally, we extend our construction to p-dimensional arrays, and in this case we obtain a -parameter distribution of , the number of p-ary trees.
... x m 1 y m 2 z m 3 The super Catalan numbers, first introduced by Ira Gessel [4], also admits the following simple formulas. ...
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
... n!(n + 1)! = 4 n Γ n + 1 2 √ πΓ(n + 2) , n ≥ 0. Sequence A000108 in the OEIS. One finds in the literature many generalizations of the Catalan numbers, such as the q-Catalan numbers [3,4], super ballot numbers [10], Fuss-Catalan numbers [5,13,21]... . Since 2014, Qi and his co-authors, in defining the Catalan-Qi numbers as a generalization of the Catalan numbers, have carried out important analytical studies of these numbers [14,15,16,17,18,19,20,21,22,23,25,26,27,29]. ...
We give some formulas and series identities involving the Pochham-mer k-symbol. In particular, we give generalizations of formulas and identities about the Catalan numbers and the Catalan-Qi numbers given in [F. Qi and B.-N. Guo, Integral representations of the Catalan numbers and their applications, Mathematics 5 (2017), no. 3, Article 40, available online at https://doi.org/10.3390/math5030040.] and [W. Chammam, Several formulas and identities related to Catalan-Qi and q-Catalan-Qi numbers, In-dian J.
... By setting r = 1 in Equation (1), it follows that P (n, 1) = C n . ...
... The first few Catalan numbers for 0 ≤ n ≤ 11 are See the sequence A000108 in the OEIS. One can find in the literature many generalizations of the Catalan-like numbers such as the q-Catalan numbers [2,3], the super ballot numbers [9], and the Fuss-Catalan numbers [4,13]. Since 2015, Qi and his co-authors introduced [1,12,19,20] the Catalan-Qi numbers as a generalization of the Catalan numbers C n and carried out important analytic studies of these numbers [5,11,14,17]. ...
In the paper, the authors establish several identities and relations involving q-analogues of the Pochhammer k-symbol. Moreover, the authors generalize several identities and relations for q-analogues of the Catalan numbers and the Catalan-Qi numbers.
... We need some preliminary work first. To this end, recall the super Catalan numbers [7] defined by ...
In this {\it case study}, we hope to show why Sheldon Axler was not just wrong, but {\em wrong}, when he urged, in 1995: ``Down with Determinants!''. We first recall how determinants are useful in enumerative combinatorics, and then illustrate three versatile tools (Dodgson's condensation, the holonomic ansatz and constant term evaluations) to operate in tandem to prove a certain intriguing determinantal formula conjectured by the first author. We conclude with a postscript describing yet another, much more efficient, method for evaluating determinants: `ask determinant-guru, Christian Krattenthaler', but advise people only to use it as a last resort, since if we would have used this last method right away, we would not have had the fun of doing it all by ourselves.
... , Ω(m, n) := S(m, n) 4 m+n and are indexed by two elements in N which for us includes 0. The super Catalan numbers were first introduced by Catalan [3] in 1874 and the first modern study of these numbers was initiated by Gessel [7] in 1992. They generalized the Catalan numbers c n since S(1, n) = 2c n . ...
We extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields of characteristic zero as a normalized -linear functional on that takes polynomials that evaluate to zero on C to zero and is -invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials .
... (2m)!/n! m! (n + m)! are always integral. Since A n,1 /2 coincides with the Catalan number C n , these A n,m were named the super Catalan numbers by Gessel [6]. The integrality of A n,m can also be deduced from Von Szily's identity [15]: ...
The integrality of the numbers was observed by Catalan as early as 1874 and Gessel named the super Catalan numbers. The positivity of the q -super Catalan numbers ( q -analogue of the super Catalan numbers) was investigated by Warnaar and Zudilin [‘A q -rious positivity’, Aequationes Math. 81 (2011), 177–183]. We prove the divisibility of sums of q -super Catalan numbers, which establishes a q -analogue of Apagodu’s congruence involving super Catalan numbers.
... (1) show that there is a close relationship between Gessel's numbers P (n, r) and super Catalan numbers S(n, r). Gessel's numbers have, at least, two combinatorial interpretations [5,9], while super Catalan numbers do not have a combinatorial interpretation in general. We think that some combinatorial properties of Gessel's numbers can be used for obtaining a combinatorial interpretation for super Catalan numbers in general. ...
The Gessel number P(n,r) is the number of the paths in plane with and (0,1) steps from (0,0) to that never touch any of the points from the set . We show that there is a close relationship between the Gessel numbers P(n,r) and the super Catalan numbers S(n,r). By using new sums, we prove that an alternating convolution of the Gessel numbers P(n,r) is always divisible by \frac{1}{2}S(n,r).
... By setting r = 1 in Equation (1), it follows that P (n, 1) = C n . ...
The Gessel number P(n,r) represents the number of lattice paths in a plane with unit horizontal and vertical steps from (0,0) to that never touch any of the points from the set . In this paper, we use combinatorial arguments to derive a recurrence relation between P(n,r) and . Also, we give a new proof for a well-known closed formula for P(n,r). Moreover, a new combinatorial interpretation for the Gessel numbers is presented.
... Computer experiments indicate that there is a family of q-numbers related to several generalizations of q-Catalan. For instance, following [Ges92] define q-super Ballot numbers ...
International audience
This paper contains two results. First, I propose a q-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These q-integers are palindromic polynomials in q with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of q-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in q-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan.
Cet document contient deux résultats. Tout d’abord, je vous propose un q-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces q-integers sont des polynômes palindromiques à q à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de q-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de q-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.
... The first few Catalan numbers for 0 ≤ n ≤ 11 are See the sequence A000108 in the OEIS. One can find in the literature many generalizations of the Catalan-like numbers such as the q-Catalan numbers [2,3], the super ballot numbers [9], and the Fuss-Catalan numbers [4,13]. Since 2015, Qi and his co-authors introduced [1,12,19,20] the Catalan-Qi numbers as a generalization of the Catalan numbers C n and carried out important analytic studies of these numbers [5,11,14,17]. ...
Except those papers listed at the site https://qifeng618.wordpress.com/2012/01/13/som-papers-jointed-with-professor-bai-ni-guo/, Professor Bai-Ni Guo also published the following papers.
... In 2018, the first author [2] conjectured two congruences on sums of the super Catalan numbers (named by Gessel [5]): ...
In this paper, we state and prove some congruence properties for the trinomial coeficients, one of which is similar to the Wolstenholme's theorem.
... The author showed that the inverse of the reciprocal Pascal matrix has integer elements which was conjectured already in [5]. For this purpose, he derived the factorization S = GM G, where the diagonal matrix G has entries G i,i = 2i i , and S is the super Catalan matrix [2,4] with entries S i,j = (2i)! (2j)! i!j! (i + j)! . ...
In this paper, we present a number of combinatorial matrices that are generalizations or variants of the super Catalan matrix and the reciprocal Pascal matrix. We present explicit formulæ for LU-decompositions of all the matrices and their inverses. Alternative derivations using hypergeometric functions are also given.
... The case m = 2 is already more interesting since it amounts to the von Szily identity [45] for super-Catalan numbers (see, e.g., [26]). Proposition 4 (von Szily) ∀a 1 , a 2 ∈ Z, ...
In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We generalize Dixmier’s collection and show that a particular subfamily is algebraically independent. Our proof relies on showing certain alternating sums of products of binomial coefficients are nonzero. Along the way we provide a very elementary proof à la Racah, namely, only using the Chu–Vandermonde Theorem, for Dixon’s Summation Theorem. We also provide explicit computations of invariants, for the binary octavic, which can serve as ideal introductory examples to Gordan’s 1868 method in classical invariant theory.
... The case m = 2 is already more interesting since it amounts to the von Szily identity [40] for super-Catalan numbers (see, e.g., [26]). ...
In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We generalize Dixmier's collection and show that a particular subfamily is algebraically independent. Our proof relies on showing certain alternating sums of products of binomial coefficients are nonzero. Along the way we provide a very elementary proof \`a la Racah, namely, only using the Chu-Vandermonde Theorem, for Dixon's Summation Theorem.
... Here, we can cite Radoux [24,25], Krattenthaler [8,9], Chammam et al. [4], and others. Also, we can refer to the work of Gessel [6] on the different combinatorial descriptions for super-Catalan numbers. Since 2014, Qi and his co-authors, in defining the Catalan-Qi function, have carried out important analytical studies of this subject (integral representations and their applications, convexity, formulas, identities, inequalities, and the like) [10,11,12,13,14,15,16,17,18,19,20,21,22,23,26]. ...
In the paper, the author generalizes several formulas and series identities involving the Catalan numbers and establishes several new formulas and series identities involving the Catalan-Qi numbers and q-Catalan-Qi numbers.
... In 1874, E. Catalan observed that the numbers S(m, n) = 2m m 2n n m+n m are integers. Since S(1, n)/2 coincides with C n , these numbers S(m, n) are named super Catalan numbers by Gessel [6]. These numbers should not be confused with the Schröder-Hipparchus numbers, which are sometimes also called super Catalan numbers. ...
... Here (A127779, A104633) are a reciprocal pair. In particular, A033820 is connected to the enumeration of paths avoiding the line x = y; see [111,202]. ...
We study linear recurrences of Eulerian type of the form with given, where and are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of for large n using the method of moments and analytic combinatorial tools under varying and , and apply our results to more than two hundred of concrete examples when and more than three hundred when that we collected from the literature and from Sloane's OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.
... For clarity, we separate the factor e n in the following Here (A127779, A104633) are a reciprocal pair. In particular, A033820 is connected to the enumeration of paths avoiding the line x D y; see [111,202]. ...
We study linear recurrences of Eulerian type of the form
P_n(v) = a_n(v) P_{n-1}(v) + b_n(v) (1-v) P_{n-1}’(v),
for n>0 with P_0(v) given, where a_n(v) and b_n(v) are in most cases
polynomials of low degrees. We characterize the various limit laws of
the coefficients of P_n(v) for large n using the method of moments
and analytic combinatorial tools under varying a_n(v) and b_n(v), and
apply our results to more than two hundred of concrete examples when
b_n(v) is not zero and more than three hundred when b_n(v)=0 that we
collected from the literature and from Sloane's OEIS database. The
limit laws and the convergence rates we worked out are almost all new
and include normal, half-normal, Rayleigh, beta, Poisson, negative
binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising
richness and diversity of such a simple framework, as well as the
power of the approaches used.
... On the other hand, our investigative process aims to introduce the Catalan numbers. Gessel (1992) discussed the validity of the following identity 1 (2 )! 64 !( 2)! nn n CC nn , for all 0 n . Based on the research that seeks to understand the numerical behavior of the expression. ...
... Indeed, they are different from the so-called super Catalan numbers by a factor 2(θ+1) θ+1 and (θ + 2)(θ + 3) · · · (2θ + 1)C θ k is an integer multiple of θ!. We refer to Gessel [16] or Hilton and Pedersen [20] for the study of this kind of generalized Catalan numbers. ...
... In 1874, E. Catalan observed that the numbers S(m, n) = 2m m 2n n m+n m are integers. Since S(1, n)/2 coincides with C n , these numbers S(m, n) are named super Catalan numbers by Gessel [5]. These numbers should not confused with the Schröder-Hipparchus numbers, which are sometimes also called super Catalan numbers. ...
In this paper, we prove two congruences on the double sums of the super Catalan numbers (named by Gessel), which were recently conjectured by Apagodu.
... (4) Gessel's [6] super ballot numbers (often also called super-Catalan numbers) (number of rooted Hamiltonian maps with 2n vertices; cf. [17]), to mention just a few. ...
We prove a central limit theorem for the joint distribution of s q ( A j n ) s_q(A_jn) , 1 ≤ j ≤ d 1\le j \le d , where s q s_q denotes the sum-of-digits function in base q q and the A j A_j ’s are positive integers relatively prime to q q . We do this in fact within the framework of quasi-additive functions. As an application, we show that most elements of “Catalan-like” sequences—by which we mean integer sequences defined by products/quotients of factorials—are divisible by any given positive integer.
... The super Catalan numbers [3] first introduced by Ira Gessel also admit the following simple formula. ...
We provide elementary proof for several congruences involving sum of binomial coefficients (single sum and multi-sum) and derive some new congruences. Also we state similar problems where our method fails to apply.
... a!b!(a+b)! [4]. Finding a combinatorial interpretation for the super-Catalan number is an open problem; when a ≤ 3 or |a − b| ≤ 3, combinatorial interpretations have been found ( [2], [3]). ...
We resolve a conjecture of Albert and Bousquet-Melou enumerating
quarter-plane walks with fixed horizontal and vertical projections according to
their upper-right-corner count modulo 2. In doing this, we introduce a signed
upper-right-corner count statistic. We find its distribution over planar walks
with any choice of fixed horizontal and vertical projections. Additionally, we
prove that the polynomial counting loops with a fixed horizontal and vertical
projection according to the absolute value of their signed upper-right-corner
count is (x+1)-positive. Finally, we conjecture an equivalence between
(x+1)-positivity of the generating function for upper-right-corner count and
signed upper-right-corner count.
The Gessel number P(n,r) is the number of lattice paths in the plane with (1,0) and (0,1) steps from (0,0) to that never touch any of the points from the set . We show that there is a close relationship between Gessel numbers P(n,r) and super Catalan numbers T(n,r). A new class of binomial sums, so called M sums, is used. By using one form of the Pfaff–Saalschütz theorem, a new recurrence relation for M sums is proved. Finally, we prove that an alternating convolution of Gessel numbers P(n,r) multiplied by a power of a binomial coefficient is always divisible by .
We give some formulas and series identities involving the Pochhammer k-symbol. In particular, we give generalizations of formulas and identities about the Catalan numbers and the Catalan–Qi numbers given in [F. Qi and B.-N. Guo, Integral representations of the Catalan numbers and their applications, Mathematics 5 (2017), no. 3, Article 40, available online at https://doi.org/10.3390/math5030040.] and [W. Chammam, Several formulas and identities related to Catalan–Qi and q-Catalan–Qi numbers, Indian J. Pure Appl Math., 50, (2019), 1039–1048; Available online at https://doi.org/10.1007/s13226-019-0372-1].
We extend the notion of polynomial integration over an arbitrary circle C in the Euclidean geometry over general fields of characteristic zero as a normalised -linear functional on that maps polynomials that evaluate to zero on C to zero and is -invariant. This allows us to not only build a purely algebraic integration theory in an elementary way, but also give the super Catalan numbers \begin{align*} S(m,n) = \frac{(2m)!(2n)!}{m!n!(m+n)!} \end{align*}
an algebraic interpretation in terms of values of this algebraic integral over some circle applied to the monomials .
We use a variant of Salikhov's ingenious proof that the irrationality measure of π is at most 7.606308... to prove that, in fact, it is at most 7.103205334137...
Registramos uma considerável atenção dedicada por parte dos autores de livros de História da Matemática (HM) concernentemente aos clássicos fundamentos do Cálculo Diferencial e Integral. Por outro lado, se mostra imprescindível ao entendimento do professor de Matemática uma compreensão sobre um irrefreável processo matemático e epistemológico evolutivo dos objetos matemáticos, desde seu estádio de nascedouro até o momento atual. Assim, o presente trabalho relata uma Engenharia Didática de Formação (EDF) desenvolvida com a participação de cinco professores em formação inicial, no Instituto Federal de Educação, Ciência e Tecnologia – IFCE, no ano de 2017. O tema abordado envolveu a noção de Números Generalizados de Catalan (NGC) que representa uma contribuição de vários matemáticos e a pesquisa atual sobre inúmeros problemas derivados confirmam seu processo de generalização ininterrupto. O estudo envolveu cinco tarefas e duas situações estruturadas de ensino, com o aporte da Teoria das Situações Didáticas (TSD). Os dados coligidos evidenciam várias propriedades e, sobretudo, teoremas e definições matemáticas descobertas e formuladas pelos sujeitos participantes da investigação o que concorreu para o incremento de suas habilidades profissionais e um conhecimento histórico, epistêmico e pragmático sobre a noção. We have recorded considerable attention on the part of the authors of Mathematical History (MH) books concerning the classical fundamentals of Differential and Integral Calculus. On the other hand, it is essential to the understanding of the Mathematics teacher an understanding about an unstoppable mathematical and evolutionary epistemological process of the mathematical objects, from its nascent stage to the present moment. Thus, the present work reports a Training Didactic Engineering (EDF) developed with the participation of five teachers in initial formation, in the Federal Institute of Education, Science and Technology - IFCE, in the year 2017. The topic covered involved the notion of Numbers Generalized Catalan (NGC) that represents a contribution of several mathematicians and the current research on numerous derived problems confirm its process of uninterrupted generalization. The study involved five tasks and two structured teaching situations, with the contribution of the Theory of Educational Situations (TSD). The collected data show several properties and, above all, the theorems and mathematical definitions discovered and formulated by the subjects participating in the research, which contributed to the increase of their professional skills and a historical, epistemic and pragmatic knowledge about the notion.
Stimulated by a remark of J. L. Doob at the beginning of Appen-dix I to Kai Lai Chung's English translation "Limit Distributions for Sums of Independent Variables" of the Russian classic by B. V. Gnedenko and A. N. Kolmogorov [5] we highlight the somewhat nonprobabilistic importance of characteristic functions and their positive definiteness property in a ped-agogical attempt to introduce the notion of a quantum Gaussian state and its properties to a classical probabilist. Such a presentation leads to some natural open problems on symmetry transformation properties of Gaussian states.
The Sun polynomials are defined by \begin{align*}
g_n(x)=\sum_{k=0}^n{n\choose k}^2{2k\choose k}x^k. \end{align*} We prove that,
for any positive integer n, there hold \begin{align*}
&\frac{1}{n}\sum_{k=0}^{n-1}(4k+3)g_k(x) \in\mathbb{Z}[x],\quad\text{and}\\
&\sum_{k=0}^{n-1}(8k^2+12k+5)g_k(-1)\equiv 0\pmod{n}. \end{align*} The first
one confirms a recent conjecture of Z.-W. Sun, while the second one partially
answers another conjecture of Z.-W. Sun. Our proof depends on the following
congruence: {m+n-2\choose m-1}{n\choose m}{2n\choose n}\equiv
0\pmod{m+n}\quad\text{for .}
Objects lying in four different boxes are rearranged in such a way that the number of objects in each box stays the same. Askey, Ismail, and Koornwinder proved that the cardinality of the set of rearrangements for which the number of objects changing boxes is even exceeds the cardinality of the set of rearrangements for which that number is odd. We give a simple counting proof of this fact. © 1983, Academic Press Inc. (London) Limited. All rights reserved.
We raise conjectures concerning the positivity of the power series coefficients of multi-variate rational functions. We also give a short proof of a result of Askey and Gasper that (1-x-y-z+4xyz) -b has positive power series coefficients for b≥(17-3)/2. We show how Ismail and Tanharkar’s proof that (1-(1-L)x-Ly-Lxz-(1-L)yz+xyz) -a (where 0≤L≤1) has positive power series coefficients for a=1 implies Koornwinder’s result that it does so for a≥1. The paper has been finally published under the same title in [SIAM J. Math. Anal. 14, 396-398 (1983; MR 84i:42017)].
p1 Present address: Trinity College, Cambridge.
Consider all words in {1,…,n}. A fixed set of words is labeled as the set of “mistakes”. A generating function for the number of words with m11's,…,mnn's and k mistakes is given. This generalizes a result of Gessel who considered the case where all the mistakes are two-lettered. A similar result has been independently obtained by Goulden and Jackson.
In this paper a very short proof is given of an identity concerning Catalan numbers due originally to Touchard.
A combinatorial approach to some positivity problems A short proof of an identity of Touchard’s concerning Catalan numbers
- M E H Ismail
- M V Tamhankar
- L W Shapiro
Ismail, M. E. H., Tamhankar, M. V. (1979), A combinatorial approach to some positivity problems, SIAM J. Math. Anal. 10, 478–485. Shapiro, L. W. (1976), A short proof of an identity of Touchard’s concerning Catalan numbers, J. Combin. Theory Ser. A 20, 375–376. v. Szily, K. (1894), Ueber die Quadratsummen der Binomialcoefficienten, Ungar. Ber. 12, 84–91. Cited in Jahrbuch ¨ uber die Fortschritte der Mathematik 25 (1893), 405–406. Zeilberger, D. (1981), Enumeration of words by their number of mistakes, Disc. Math. 34, 89–91. 16
- K Szily
Szily, K. (1894), Ueber die Quadratsummen der Binomialcoefficienten, Ungar. Ber. 12, 84–91. Cited
in JahrbuchüberJahrbuch¨Jahrbuchüber die Fortschritte der Mathematik 25 (1893), 405–406.
- R Askey
Askey, R. (1975), Orthogonal Polynomials and Special Functions, CBMS Regional Conference Series
in Applied Mathematics, Vol. 21. Philadelphia: Society for Industrial and Applied Mathematics.
Ueber die Quadratsummen der Binomialcoefficienten
- v. Szily