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A001263
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Triangle of Narayana numbers T(n,k) = C(n-1,k-1)C(n,k-1)/k with 1<=k<=n, read by rows. Also called the Catalan triangle.
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214
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1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 20, 10, 1, 1, 15, 50, 50, 15, 1, 1, 21, 105, 175, 105, 21, 1, 1, 28, 196, 490, 490, 196, 28, 1, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 1, 55, 825, 4950, 13860, 19404, 13860, 4950, 825
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OFFSET
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1,5
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COMMENTS
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Antichains (or order ideals) in the poset 2*(k-1)*(n-k) or plane partitions with rows <= k-1, columns <= n-k and entries <= 2 - Mitch Harris, Jul 15 2000
a(n,k) = number of Dyck n-paths with exactly k peaks. a(n,k) = number of pairs (P,Q) of lattice paths from (0,0) to (k,n+1-k), each consisting of unit steps East or North, such that P lies strictly above Q except at the endpoints. - David Callan, Mar 23 2004
Number of permutations of [n] which avoid-132 and have k-1 descents - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004
a(n,k) is the number of paths through n panes of glass, entering and leaving from one side, of length 2n with k reflections (where traversing one pane of glass is the unit length). - Mitch Harris, Jul 06 2006
Antidiagonal sums given by A004148 (without first term).
T(n,k) is the number of full binary trees with n internal nodes and k-1 jumps. 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. - Emeric Deutsch, Jan 18 2007
A108679 equals the 6th column of the table of Narayana numbers - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 18 2007
Comment from Gary W. Adamson, Oct 22 2007: (Start) The n-th row can be generated by the following operation using an ascending row of (n-1) triangular terms, (A) and a descending row, (B); e.g. row 6:
A: 1....3....6....10....15
B: 15...10....6.....3.....1
C: 1...15...50....50....15....1 = row 6.
(Leftmost column of A,B -> first two terms of C; then followed by the operation B*C/A of current column = next term of row C, (e.g. 10*15/3 = 50). Continuing with the operation, we get row 6: (1, 15, 50, 50, 15, 1). (End)
The previous comment can be upgraded to: The ConvOffsStoT transform of the triangular series; and by rows, row 6 is the ConvOffs transform of (1, 3, 6, 10, 15). Refer to triangle A117401 as another example of the ConvOffsStoT transform, and OEIS under Maple Transforms. - Gary W. Adamson, Jul 09 2012
Other versions are in A090181 and A131198 . - Philippe DELEHAM, Nov 18 2007
From the Kostov-Shapiro paper: In the present paper we find a new interpretation of Narayana polynomials N_n(x) which are the generating polynomials for the Narayana numbers N_{n,k} counting Dyck paths of length n and with exactly k peaks. Strangely enough Narayana polynomials also occur as limits as n->oo of the sequences of eigenpolynomials of the Schur-Szego composition map sending (n-1)-tuples of polynomials of the form (x+1)^{n-1}(x+a) to their Schur-Szego product, see below. As a corollary we obtain that every N_n(x) has all roots real and non-positive. Additionally, we present an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {N_n(x)}. - Jonathan Vos Post, Apr 08 2008
For a connection to Lagrange inversion, see A134264. [From Tom Copeland, Aug 15 2008]
T(n,k) is also the number of order-decreasing and order-preserving mappings (of an n-element set) of height k (height of a mapping is the cardinal of its image set). [From Abdullahi Umar, Aug 21 2008]
Contribution from Peter Bala, Oct 27 2008: (Start)
Row n of this triangle is the h-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p.60]. See A033282 for the corresponding array of f-vectors for associahedra of type A_n . See A008459 and A145903 for the h-vectors for associahedra of type B and type D respectively. The Hilbert transform of this triangle (see A145905 for the definition of this transform) is A145904.
(End)
T(n,k) is also the number of noncrossing set partitions of [n] into k blocks. Given a partition P of the set {1,2,...,n}, a crossing in P are four integers [a, b, c, d] with 1 <= a < b < c < d <= n for which a, c are together in a block, and b, d are together in a different block. A noncrossing partition is a partition with no crossings. - Peter Luschny, Apr 29 2011
Diagonals of A089732 are rows of A001263.- Tom Copeland, May 14 2012
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REFERENCES
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M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.- From N. J. A. Sloane, Oct 08 2012
Benchekroun, S.; Moszkowski, P.; A bijective proof of an enumerative property of legal bracketings. Discrete Math. 176 (1997), no. 1-3, 273-277.
Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
M. Bona and B. E. Sagan, On Divisibility of Narayana Numbers by Primes, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.4.
Carlitz, L.; Riordan, John. Enumeration of some two-line arrays by extent. J. Combinatorial Theory Ser. A 10 1971 271--283. MR0274301(43 #66). (Coefficients of the polynomials A_n(z) defined in (3.9)). - N. J. A. Sloane, May 15 2012
Yun Ding and Rosena R. X. Du, Counting Humps in Motzkin paths, Arxiv preprint arXiv:1109.2661, 2011
T. Doslic, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discr. Math., 285 (2004), 67-82.
A. Frabetti, Simplicial properties of the set of planar binary trees, J. Algebraic Combin., 13 (2001), 41-65.
S. Giraudo, Intervals of balanced binary trees in the Tamari lattice, Arxiv preprint arXiv:1107.3472, 2011
Graham, R. L. and J. Riordan, The solution of a certain recurrence, Amer. Math. Monthly 73, 1966, pp. 604-608 [From Roger L. Bagula, Feb 13 2009]
H. E. Hoggatt, Jr., Triangular Numbers, Fibonacci Quarterly 12 (Oct. 1974), 221-230.
Hwang, F. K.; Mallows, C. L.; Enumerating nested and consecutive partitions. J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, Arxiv preprint arXiv:1208.1052, 2012. - From N. J. A. Sloane, Dec 24 2012
W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.
G. Kreweras, Sur les e'ventails de segments, {\em Cahiers du Bureau Universitaire de Recherche Op\'{e}rationnelle}, Institut de Statistique, Universit\'{e} de Paris, #15 (1970), 3-41.
G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
Laradji, A. and Umar, A. On certain finite semigroups of order-decreasing transformations I, Semigroup Forum, 69 (2004), 184-200. [From Abdullahi Umar, Aug 21 2008]
P. A. MacMahon, Combinatory Analysis, Sect. 495, 1916.
Toufik Mansour and Mark Shattuck, Pattern-avoiding set partitions and Catalan numbers, Electronic Journal of Combinatorics, 18(2) (2012), #P34. From N. J. A. Sloane, Jun 15 2012
T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
J. Riordan, Combinatorial Identities, Wiley, 1968, p.17 [From Roger L. Bagula, Feb 13 2009]
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.36(a) and (b).
F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.
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LINKS
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T. D. Noe, Rows n=1..100 of triangle, flattened
M. Bona and B. E. Sagan, On divisibility of Narayana numbers by primes
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 182
S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004. [From Peter Bala, Oct 27 2008]
Alessandra Frabetti, Simplicial properties of the set of planar binary trees.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6 [From Peter Bala, Oct 22 2008]
F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras
Thomas Koshy, Illustration of triangle with dark color for odd number, light for even number [Although the illustration says "Applet", this is simply a plain jpeg file]
Vladimir Kostov and Boris Shapiro, Narayana numbers and Schur-Szego composition
A. Laradji and A. Umar, Combinatorial Results for Semigroups of Order-Decreasing Partial Transformations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.8.
P. A. MacMahon, Combinatory analysis.
D. Merlini, R. Sprugnoli and M. C. Verri, Waiting patterns for a printer, FUN with algorithm'01, Isola d'Elba, 2001.
A. Micheli and D. Rossin, Edit distance between unlabeled ordered trees
J.-C. Novelli and J.-Y. Thibon, Polynomial realizations of some trialgebras, Proc. Formal Power Series and Algebraic Combinatorics 2006 (San-Diego, 2006)
J.-C. Novelli and J.-Y. Thibon, Hopf algebras and dendriform structures arising from parking functions, Fundamenta Mathematicae 193 (2007), pp. 189-241, arXiv:0511200.
R. A. Sulanke, Moments, Narayana numbers and the cut and paste for lattice paths
R. A. Sulanke, Three-dimensional Narayana and Schr\"oder numbers
Wikipedia, Noncrossing partition
L. K. Williams, Enumeration of totally positive Grassmann cells
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FORMULA
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a(n, k) = C(n-1, k-1)C(n, k-1)/k for k!=0; a(n, 0)=0.
Triangle equals [0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is Deleham's operator defined in A084938.
0<n, 1<=k<=n a(n, 1) = a(n, n) = 1 a(n, k) = sum(i=1..n-1, sum(r=1..k-1, a(n-1-i, k-r) a(i, r))) + a(n-1, k) a(n, k) = sum(i=1..k-1, binomial(n+i-1, 2k-2)*a(k-1, i)) - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Aug 26 2004
T(n, k) = C(n, k)*C(n-1, k-1) - C(n, k-1)*C(n-1, k) (determinant of a 2 X 2 subarray of Pascal's triangle A007318) - Gerald McGarvey, Feb 24 2005
T(n, k) = binomial(n-1, k-1)^2 - binomial(n-1, k)binomial(n-1, k-2). - David Callan, Nov 02 2005
a(n,k) = C(n,2) (a(n-1,k)/((n-k)(n-k+1)) + a(n-1,k-1)/(k(k-1))) a(n,k) = C(n,k) C(n,k-1)/n - Mitch Harris, Jul 06 2006
G.f.: (1-x*(1+y)-sqrt((1-x*(1+y))^2-4*y*x^2))/(2*x) = Sum_{n>0, k>0} a(n, k)*x^n*y^k.
Contribution from Peter Bala, Oct 22 2008: (Start)
Relation with Jacobi polynomials of parameter (1,1):
Row n+1 generating polynomial equals 1/(n+1)*x*(1-x)^n*Jacobi_P(n,1,1,(1+x)/(1-x)). It follows that the zeros of the Narayana polynomials are all real and non-positive, as noted above. O.g.f for column k+2: 1/(k+1) * y^(k+2)/(1-y)^(k+3) * Jacobi_P(k,1,1,(1+y)/(1-y)). Cf. A008459.
T(n+1,k) is the number of walks of n unit steps on the square lattice (i.e. each step in the direction either up (U), down (D), right (R) or left (L)) starting from the origin and finishing at lattice points on the x axis and which remain in the upper half-plane y >= 0 [Guy]. For example, T(4,3) = 6 counts the six walks RRL, LRR, RLR, UDL, URD and RUD, from the origin to the lattice point (1,0), each of 3 steps. Compare with tables A145596 - A145599.
Define a functional I on formal power series of the form f(x) = 1 + ax + bx^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim n -> infinity f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).
The o.g.f. for this array is I(1 + t*x + t*x^2 + t*x^3 + ...) = 1 + t*x + (t + t^2)*x^2 + (t + 3*t^2 + t^3)*x^3 + ... = 1/(1 - x*t/(1 - x/(1 - x*t/(1 - x/(1 - ...))))) (as a continued fraction). Cf. A108767, A132081 and A141618.
(End)
Contribution from Paul Barry, Sep 28 2010: (Start)
G.f.: 1/(1-x-xy-x^2y/(1-x-xy-x^2y/(1-... (continued fraction).
E.g.f.: exp((1+y)x)Bessel_I(1,2*sqrt(y)x)/(sqrt(y)*x). (End)
G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n ). [From Paul D. Hanna, Oct 13 2010]
Contribution from Tom Copeland, Sept 04 2011: (Start)
With F(x,t) = (1-(1+t)*x-sqrt(1-2*(1+t)*x+((t-1)*x)^2))/(2*x) an o.g.f. in x for the Narayana polynomials in t, G(x,t) = x/(t+(1+t)*x+x^2) is the compositional inverse in x.
Consequently, with H(x,t) = 1/ (dG(x,t)/dx) = (t+(1+t)*x+x^2)^2 / (t-x^2), the n-th Narayana polynomial in t is given by (1/n!)*((H(x,t)*D_x)^n)x evaluated at x=0, i.e., F(x,t) = exp(x*H(u,t)*D_u)u, evaluated at u = 0. Also, dF(x,t)/dx = H(F(x,t),t). (End)
Contribution from Tom Copeland, Sept 21 2011: (Start)
With offset 0, A001263 = sum(j=0,1,...,infinity) A132710^j / A010790(j), a normalized Bessel fct. May be represented as the Pascal matrix A007318,
n!/[(n-k)!*k!], umbralized with b(n)=A002378(n) for n>0 and b(0)=1:
A001263(n,k)= b.(n!)/{b.[(n-k)!]*b.(k!)} where b.(n!) = b(n)*b(n-1)...*b(0), a generalized factorial (see example). (End)
Contribution from Tom Copeland, Sept 30 2011: (Start)
With F(x,t) = {1-(1-t)*x-sqrt[1-2*(1+t)*x+[(t-1)*x]^2]}/2 a shifted o.g.f. in x for the Narayana polynomials in t, G(x,t)= x/[t-1+1/(1-x)] is the compositional inverse in x.
Therefore, with H(x,t)=1/(dG(x,t)/dx)=[t-1+1/(1-x)]^2/{t-[x/(1-x)]^2}, (see A119900), the (n-1)-th Narayana polynomial in t is given by
(1/n!)*((H(x,t)*d/dx)^n)x evaluated at x=0, i.e.,
F(x,t) = exp(x*H(u,t)*d/du) u, evaluated at u = 0.
Also, dF(x,t)/dx = H(F(x,t),t). (End)
T(n,k)=binomial(n-1,k-1)*binomial(n+1,k)-binomial(n,k-1)*binomial(n,k). - From Philippe Deléham, Nov 05 2011
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EXAMPLE
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For all n are 12...n (1 block) and 1|2|3|...|n (n blocks) noncrossing set partitions.
[1] 1
[2] 1, 1
[3] 1, 3, 1
[4] 1, 6, 6, 1
[5] 1, 10, 20, 10, 1
[6] 1, 15, 50, 50, 15, 1
Example of umbral representation:
A007318(5,k)=[1,5/1,5*4/(2*1),...,1]=(1,5,10,10,5,1),
so A001263(5,k)={1,b(5)/b(1),b(5)*b(4)/[b(2)*b(1)],...,1}
= [1,30/2,30*20/(6*2),...,1]=(1,15,50,50,15,1).
First = last term = b.(5!)/[b.(0!)*b.(5!)]= 1.- Tom Copeland, Sep 21 2011
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MAPLE
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a := (n, k)->binomial(n-1, k-1)*binomial(n, k-1)/k;
a:=proc(n, k) option remember; local i; if k=1 or k=n then 1 else add(binomial(n+i-1, 2*k-2)*a(k-1, i), i=1..k-1); fi; end:
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MATHEMATICA
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T[n_, k_] := If[k==0, 0, Binomial[n-1, k-1]Binomial[n, k-1]/k]
Flatten[Table[Binomial[n-1, k-1] Binomial[n, k-1]/k, {n, 15}, {k, n}]] (* From Harvey P. Dale, Feb 29 2012 *)
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PROG
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(PARI) a(n, k)=if(k==0, 0, binomial(n-1, k-1)*binomial(n, k-1)/k)
(PARI) {T(n, k)=polcoeff(polcoeff(exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*y^j)*x^m/m)+O
(x^(n+1))), n, x), k, y)} [From Paul D. Hanna, Oct 13 2010]
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CROSSREFS
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Cf. A000372, A002083, A056932, A056939, A056940, A056941, A065329, A073345.
Columns give A000217, A002415, A006542, A006857, A084938.
Row sums gives A000108 Catalan numbers, n>0.
Central column = A000891, (2n)!(2n+1)! / (n! (n+1)!)^2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 29 2006.
Cf. A090181 (triangle with additional leading (trivial) row and column).
A008459, A145596, A145597, A145598, A145599. [From Peter Bala, Oct 22 2008]
A008459 (h-vectors type B associahedra), A033282 (f-vectors type A associahedra), A145903 (h-vectors type D associahedra), A145904 (Hilbert transform). [From Peter Bala, Oct 27 2008]
Cf. variants: A181143, A181144. [From Paul D. Hanna, Oct 13 2010]
Cf. A016098 and A189232 for numbers of crossing set partitions.
Sequence in context: A114176 A056241 A162745 * A162747 A107105 A088925
Adjacent sequences: A001260 A001261 A001262 * A001264 A001265 A001266
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KEYWORD
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nonn,easy,tabl,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Updated reference Charles R Greathouse IV, Oct 28 2009
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STATUS
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approved
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