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A009766 Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j). 97
1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 9, 14, 14, 1, 5, 14, 28, 42, 42, 1, 6, 20, 48, 90, 132, 132, 1, 7, 27, 75, 165, 297, 429, 429, 1, 8, 35, 110, 275, 572, 1001, 1430, 1430, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

The entries in this triangle (in its many forms) are often called ballot numbers.

T(n,k) = number of standard tableaux of shape (n,k) (n > 0, 0 <= k <= n). Example: T(3,1) = 3 because we have 134/2, 124/3 and 123/4. - Emeric Deutsch, May 18 2004

T(n,k) is the number of full binary trees with n+1 internal nodes and jump-length k. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given ordered tree is called the jump-length. - Emeric Deutsch, Jan 18 2007

The k-th diagonal from the right (k=1, 2, ...) gives the sequence obtained by asking in how many ways can we toss a fair coin until we first get k more heads than tails. The k-th diagonal has formula k(2m+k-1)!/(m!(m+k)!) and g.f. (C(x))^k where C(x) is the generating function for the Catalan numbers, (1-sqrt(1-4*x))/(2*x). - Anthony C Robin, Jul 12 2007

T(n,k) is also the number of order-decreasing and order-preserving full transformations (of an n-element chain) of waist k (waist (alpha) = max(Im(alpha))). - Abdullahi Umar, Aug 25 2008

Formatted as an upper right triangle (see tables) a(c,r) is the number of different triangulated planar polygons with c+2 vertices, with triangle degree c-r+1 for the same vertex X (c=column number, r=row number, with c>=r>=1). - Patrick Labarque, Jul 28 2010

The triangle sums, see A180662 for their definitions, link Catalan's triangle, its mirror is A033184, with several sequences, see crossrefs. - Johannes W. Meijer, Sep 22 2010

The m-th row of Catalan's triangle consists of the unique nonnegative differences of the form binomial(m+k,m)-binomial(m+k,m+1) with 0<=k<=m (See Links). - R. J. Cano, Jul 22 2014

T(n,k) is also the number of nondecreasing parking functions of length n+1 whose maximum element is k+1. For example T(3,2) = 5 because we have (1,1,1,3), (1,1,2,3), (1,2,2,3), (1,1,3,3), (1,2,3,3). - Ran Pan, Nov 16 2015

REFERENCES

William Feller,  "Introduction to Probability Theory and its Applications", vol. I, ed. 2, chap.3, sect.1,2.

J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.

Ki Hang Kim, Douglas G. Rogers, and Fred W. Roush, Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577-594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013).

D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 22, p. 451.

C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146.

Andrzej Proskurowski and Ekaputra Laiman, Fast enumeration, ranking, and unranking of binary trees. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982). Congr. Numer. 35 (1982), 401-413.MR0725898 (85a:68152).

LINKS

T. D. Noe, Rows n=0..100 of triangle, flattened

J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.

Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, arXiv:0711.0906 [math.CO], 2007.

Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, Discrete Math., 308 (2008), 4660-4669

Axel Bacher, Antonio Bernini, Luca Ferrari, Benjamin Gunby, Renzo Pinzani, Julian West, The Dyck pattern poset Discrete Math. 321 (2014), 12--23. MR3154009.

D. F. Bailey, Counting arrangements of 1's and-1's, Mathematics Magazine, 69 (1996): 128-131. See table on p. 129.

Elena Barcucci, Alberto Del Lungo, Elisa Pergola, Renzo Pinzani, A methodology for plane tree enumeration, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995). Discrete Math. 180 (1998), no. 1-3, 45--64. MR1603693 (98m:05090).

E. Barcucci and Verri, Some more properties of Catalan numbers, Discrete Math., 102 (1992), 229-237.

Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.

A. Bernini, L. Ferrari, R. Pinzani and J. West, The Dyck pattern poset, arXiv preprint arXiv:1303.3785 [math.CO], 2013.

N. Borie, Combinatorics of simple marked mesh patterns in 132-avoiding permutations, arXiv preprint arXiv:1311.6292 [math.CO], 2013.

M. Bousquet-Mélou and M. Petkovsek, Linear recurrences with constant coefficients: the multivariate case, Discrete Math. 225 (2000), 51-75.

E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.

S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.

R. J. Cano, Catalan's books

L. Carlitz, Sequences, paths, ballot numbers

I. J. Dejter, A new approach to the middle levels via a Catalan-number system of numeration, 2015.

Italo J. Dejter, A numeral system for the middle levels, preprint, 2014. [See Section 2. - N. J. A. Sloane, Apr 06 2014]

Italo J. Dejter, Dihedral-symmetry middle-levels problem via a Catalan system of numeration, preprint, 2015.

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672. (Y_{N}(K) = A009766(N+1,K-1), 1 <= K <= N+1, N >=0 if alpha = 1 = beta).

E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.

Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.

Paul Drube, Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers, arXiv:1606.04869 [math.CO], 2016.

R. Ehrenborg, S. Kitaev, E. Steingrimsson, Number of cycles in the graph of 312-avoiding permutations, arXiv preprint arXiv:1310.1520 [math.CO], 2013.

W. J. R. Eplett, A note about the Catalan triangle, Discrete Math. 25(1979), no. 3, 289--291. MR0534947 (80i:05007)

G. Feinberg, K.-H. Lee, Homogeneous representations of KLR-algebras and fully commutative elements, arXiv preprint arXiv:1401.0845 [math.RT], 2014.

I. Fanti, A. Frosini, E. Grazzini, R. Pinzani and S. Rinaldi, Characterization and enumeration of some classes of permutominoes, PU. M. A., Vol. 18 (2007), No. 3-4, pp. 265-290.

D. Foata, G-N. Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126

Dominique Foata and Guo-Niu Han, Doubloons and new q-tangent numbers, Quart. J. Math. 62 (2) (2011) 417-432

H. G. Forder, Some problems in combinatorics, Math. Gazette, vol. 45, 1961, 199-201.

C. A. Francisco, J. Mermin, J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, 2013.

Ira Gessel, Super ballot numbers.

Niket Gowravaram, A Variation of the nil-Temperley-Lieb algebras of type A, Preprint, 2015.

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

F. Hivert, J.-C. Novelli and J.-Y. Thibon, The Algebra of Binary Search Trees, Theoretical Computer Science, 339 (2005), 129-165.

R. L. Hudson, Y. Pei, On a causal quantum stochastic double product integral related to Lévy area, Research Gate, 2015.

R. L. Hudson, Y. Pei, On a quantum causal stochastic double product integral related to Levy area, arXiv preprint arXiv:1506.04294 [math-ph], 2015.

A. Karttunen, Some notes on Catalan's Triangle.

W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.

G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.

G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]

C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146. [Annotated scanned copy]

A. Laradji and A. Umar, On certain finite semigroups of order-decreasing transformations I, Semigroup Forum 69 (2004), 184-200.

D. Merlini et al., Underdiagonal lattice paths with unrestricted steps, Discrete Appl. Math., 91 (1999), 197-213.

D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table I).

Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015.

J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.

M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014.

P. Pagacz, M. Wojtylak, On the spectral properties of a class of H-selfadjoint random matrices and the underlying combinatorics, arXiv preprint arXiv:1310.2122 [math.PR], 2013.

R. Parviainen, Permutations, cycles and the pattern 2-13, arXiv:math/0607793 [math.CO], 2006.

R. Pemantle and M. C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 261

Planet Math, Lattice Paths and ballot numbers.

L. Pudwell, Avoiding an Ordered Partition of Length 3, 2013.

C. M. Ringel, The Catalan combinatorics of the hereditary artin algebras, arXiv preprint arXiv:1502.06553 [math.RT], 2015.

A. Robertson, D. Saracino and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.

L. W. Shapiro, A Catalan triangle, Discrete Math., 14, 83-90, 1976.

T. Sillke, Catalan's numbers

R. A. Sulanke, Guessing, ballot numbers and refining Pascal's triangle

Yidong Sun, A simple bijection between binary trees and colored ternary trees, El. J. Combinat. 17 (2010) #N20

Eric Weisstein's World of Mathematics, Catalan's Triangle

Eric Weisstein's World of Mathematics, Nonnegative Partial Sum

FORMULA

a(n, m) = binomial(n+m, n)*(n-m+1)/(n+1), 0 <= m <= n.

G.f. for column m: (x^m)*N(2; m-1, x)/(1-x)^(m+1), m >= 0, with the row polynomials from triangle A062991 and N(2; -1, x) := 1.

G.f. C(t*x)/(1-x*C(t*x)) = 1+(1+t)*x+(1+2*t+2*t^2)*x^2+..., where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function. - Emeric Deutsch, May 18 2004

Another version of triangle T(n, k) given by [1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 1, 1, 1, 1, 1, ...] = 1; 1, 0; 1, 1, 0; 1, 2, 2, 0; 1, 3, 5, 5, 0; 1, 4, 9, 14, 14, 0; ...where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 16 2005

O.g.f. (with offset 1) is the series reversion of x*(1+x*(1-t))/(1+x)^2 = x - x^2*(1+t) + x^3*(1+2*t) - x^4*(1+3*t) + ... . - Peter Bala, Jul 15 2012

Sum_{k=0..floor(n/2)} T(n-k+p-1, k+p-1) = A001405(n+2*p-2) - C(n+2*p-2, p-2), p >= 1. - Johannes W. Meijer, Oct 03 2013

Let A(x,t) denote the o.g.f. Then 1 + x*d/dx(A(x,t))/A(x,t) = 1 + (1 + t)*x + (1 + 2*t + 3*t^2)*x^2 + (1 + 3*t + 6*t^2 + 10*t^3)*x^3 + ... is the o.g.f. for A059481. - Peter Bala, Jul 21 2015

EXAMPLE

Triangle begins in row n=0 with 0<=k<=n:

1

1 1

1 2  2

1 3  5   5

1 4  9  14  14

1 5 14  28  42   42

1 6 20  48  90  132  132

1 7 27  75 165  297  429  429

1 8 35 110 275  572 1001 1430 1430

1 9 44 154 429 1001 2002 3432 4862 4862

MAPLE

A009766 := proc(n, k) binomial(n+k, n)*(n-k+1)/(n+1); end proc:

seq(seq(A009766(n, k), k=0..n), n=0..10); # R. J. Mathar, Dec 03 2010

MATHEMATICA

Flatten[NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* Birkas Gyorgy, May 19 2012 *)

T[n_, k_] := T[n, k] = Which[k == 0, 1, k>n, 0, True, T[n-1, k] + T[n, k-1] ]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 07 2016 *)

PROG

(PARI) {T(n, k) = if( k<0 || k>n, 0, binomial(n+1+k, k) * (n+1-k) / (n+1+k) )}; /* Michael Somos, Oct 17 2006 */

(PARI) b009766=(n1=0, n2=100)->{my(q=if(!n1, 0, binomial(n1+1, 2))); for(w=n1, n2, for(k=0, w, write("b009766.txt", 1*q" "1*(binomial(w+k, w)-binomial(w+k, w+1))); q++))} \\ R. J. Cano, Jul 22 2014

(Haskell)

a009766 n k = a009766_tabl !! n !! k

a009766_row n = a009766_tabl !! n

a009766_tabl = iterate (\row -> scanl1 (+) (row ++ [0])) [1]

-- Reinhard Zumkeller, Jul 12 2012

(Sage)

@CachedFunction

def ballot(p, q):

    if p == 0 and q == 0: return 1

    if p < 0 or p > q: return 0

    S = ballot(p-2, q) + ballot(p, q-2)

    if q % 2 == 1: S += ballot(p-1, q-1)

    return S

A009766 = lambda n, k: ballot(2*k, 2*n)

for n in (0..7): [A009766(n, k) for k in (0..n)]  # Peter Luschny, Mar 05 2014

CROSSREFS

The following are all versions of (essentially) the same Catalan triangle: A009766, A008315, A028364, A030237, A047072, A053121, A059365, A062103, A099039, A106566, A130020, A140344.

Cf. A062745, A214292.

Sums of the shallow diagonals give A001405, central binomial coefficients: 1=1, 1=1, 1+1=2, 1+2=3, 1+3+2=6, 1+4+5=10, 1+5+9+5=20, ...

Row sums as well as the sums of the squares of the shallow diagonals give Catalan numbers (A000108).

Reflected version of A033184.

Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

Triangle sums (see the comments): A000108 (Row1), A000957 (Row2), A001405 (Kn11), A014495 (Kn12), A194124 (Kn13), A030238 (Kn21), A000984 (Kn4), A000958 (Fi2), A165407 (Ca1), A026726 (Ca4), A081696 (Ze2).

Sequence in context: A188181 A064581 A064580 * A059718 A076038 A095788

Adjacent sequences:  A009763 A009764 A009765 * A009767 A009768 A009769

KEYWORD

nonn,tabl,nice

AUTHOR

Wouter Meeussen

STATUS

approved

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Last modified September 27 03:56 EDT 2016. Contains 276590 sequences.