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A001263 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. 219
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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, 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

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

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. - 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). - Abdullahi Umar, Aug 21 2008

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. - Peter Bala, Oct 27 2008

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

From Peter Bala, Aug 07 2013: (Start)

Let E(y) = sum {n >= 0} y^n/(n!*(n+1)!) = 1/sqrt(y)*BesselI(1,2*sqrt(y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!*(n+1)! as defined in Wang and Wang.

Generating function E(y)*E(x*y) = 1 + (1 + x)*y/(1!*2!) + (1 + 3*x + x^2)*y^2/(2!*3!) + (1 + 6*x + 6*x^2 + x^3)*y^3/(3!*4!) + .... Cf. A105278 with a generating function exp(y)*E(x*y).

The n-th power of this array has a generating function E(y)^n*E(x*y). In particular, the matrix inverse A103364 has a generating function E(x*y)/E(y). (End)

A166360(n-k) = T(n,k) mod 2. - Reinhard Zumkeller, Oct 10 2013

T(n,k) is the number of nonintersecting n arches above the x axis, starting and ending on vertices 1 to 2n, with k being the number of arches starting on an odd vertice and ending on a higher even vertice. Example: T(3,2)=3  [16,25,34] [14,23,56] [12,36,45]. - Roger Ford, Jun 14 2014

REFERENCES

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

Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124

J. Cigler, Some remarks on lattice paths in strips along the x-axis; http://homepage.univie.ac.at/johann.cigler/preprints/lattice-paths.pdf, 2014.

R. Cori, G. Hetyei, How to count genus one partitions, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344; http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/viewFile/dmAT0130/4488

R. L. Graham and J. Riordan, The solution of a certain recurrence, Amer. Math. Monthly 73, 1966, pp. 604-608.

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.

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 Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41.

G. Kreweras, Les préordres 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.

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.

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.

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.

LINKS

T. D. Noe, Rows n=1..100 of triangle, flattened

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.

S. Benchekroun, P. Moszkowski, A bijective proof of an enumerative property of legal bracketings Discrete Math. 176 (1997), no. 1-3, 273-277.

Carl M. Bender and Gerald V. Dunne, Polynomials and operator orderings, J. Math. Phys. 29 (1988), 1727-1731;

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

M. Bona and B. E. Sagan, On divisibility of Narayana numbers by primes

M. Bona and B. E. Sagan, On Divisibility of Narayana Numbers by Primes, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.4.

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

D. Callan, T. Mansour, M. Shattuck, Restricted ascent sequences and Catalan numbers, arXiv preprint arXiv:1403.6933, 2014

L. Carlitz, and John Riordan, 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)).

R. Cori, G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628, 2013

R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv preprint arXiv:1310.2449, 2013

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.

FindStat - Combinatorial Statistic Finder, The number of internal nodes of an ordered tree

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.

Alessandra Frabetti, Simplicial properties of the set of planar binary trees.

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

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6

F. Hivert, J.-C. Novelli and J.-Y. Thibon, Commutative combinatorial Hopf algebras

M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052, 2012.

S. Kamioka, Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds, arXiv preprint arXiv:1309.0268, 2013

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.

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014. See Fig. 4.

R. A. Sulanke, Moments, Narayana numbers and the cut and paste for lattice paths

R. A. Sulanke, Three-dimensional Narayana and Schröder numbers

W. Wang and T. Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Wikipedia, Noncrossing partition

L. K. Williams, Enumeration of totally positive Grassmann cells

J. Wuttke, The zig-zag walk with scattering and absorption on the real half line and in a lattice model, J. Phys. A 47 (2014), 215203, 1-9.

FORMULA

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 Deléham'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, 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

Central column = A000891, (2n)!(2n+1)! / (n! (n+1)!)^2. - Zerinvary Lajos, Oct 29 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.

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)

G.f.: 1/(1-x-xy-x^2y/(1-x-xy-x^2y/(1-... (continued fraction). - Paul Barry, Sep 28 2010

E.g.f.: exp((1+y)x)Bessel_I(1,2*sqrt(y)x)/(sqrt(y)*x). - Paul Barry, Sep 28 2010

G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n ). - Paul D. Hanna, Oct 13 2010

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). - Tom Copeland, Sep 04 2011

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). - Tom Copeland, Sep 21 2011

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). - Tom Copeland, Sep 30 2011

T(n,k) = binomial(n-1,k-1)*binomial(n+1,k)-binomial(n,k-1)*binomial(n,k). - Philippe Deléham, Nov 05 2011

Damped sum of a column, in leading order: Lim_{d->0} d^(2k-1) Sum_{N>=k} T(N,k)(1-d)^N=Catalan(n). - Joachim Wuttke, Sep 11 2014

EXAMPLE

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

[7] 1, 21, 105,  175,  105,   21,   1

[8] 1, 28, 196,  490,  490,  196,  28,  1

[9] 1, 36, 336, 1176, 1764, 1176, 336, 36, 1; etc.

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

MAPLE

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:

MATHEMATICA

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}]] (* Harvey P. Dale, Feb 29 2012 *)

PROG

(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)} \\ Paul D. Hanna, Oct 13 2010

(Haskell)

a001263 n k = a001263_tabl !! (n-1) !! (k-1)

a001263_row n = a001263_tabl !! (n-1)

a001263_tabl = zipWith dt a007318_tabl (tail a007318_tabl) where

   dt us vs = zipWith (-) (zipWith (*) us (tail vs))

                          (zipWith (*) (tail us ++ [0]) (init vs))

-- Reinhard Zumkeller, Oct 10 2013

CROSSREFS

Other versions are in A090181 and A131198. - Philippe Deléham, Nov 18 2007

Cf. variants: A181143, A181144. - Paul D. Hanna, Oct 13 2010

Row sums give A000108 (Catalan numbers), n>0.

Columns give A000217, A002415, A006542, A006857, A084938.

A108679 (6th column). - Zerinvary Lajos, Jun 18 2007

Cf. A000372, A002083, A056932, A056939, A056940, A056941, A065329, A073345.

A145596, A145597, A145598, A145599. - 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). - Peter Bala, Oct 27 2008

Cf. A016098 and A189232 for numbers of crossing set partitions.

Cf. A243752.

Sequence in context: A114176 A056241 A162745 * A162747 A107105 A088925

Adjacent sequences:  A001260 A001261 A001262 * A001264 A001265 A001266

KEYWORD

nonn,easy,tabl,nice,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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