A056939 Array read by antidiagonals: number of antichains (or order ideals) in the poset 3*m*n or plane partitions with rows <= m, columns <= n and entries <= 3.

1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 20, 50, 20, 1, 1, 35, 175, 175, 35, 1, 1, 56, 490, 980, 490, 56, 1, 1, 84, 1176, 4116, 4116, 1176, 84, 1, 1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1, 1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1

Offset: 0

Author

Mitch Harris

Keywords

• nonn
• easy
• tabl
• nice

Comments

Triangle of generalized binomial coefficients (n,k)A342889.%20This%20array%20is%20the%20main%20subject%20of%20the%20long%20article%20by%20Felsner%20et%20al.%20(2011).%20-%20_N.%20J.%20A.%20Sloane">3; cf. A342889. This array is the main subject of the long article by Felsner et al. (2011). - _N. J. A. Sloane, Apr 03 2021

This triangle is mentioned by Hoggatt (1977). - N. J. A. Sloane, Mar 27 2021

Determinants of 3 X 3 subarrays of Pascal's triangle A007318 (a matrix entry being set to 0 when not present). - Gerald McGarvey, Feb 24 2005

Also determinants of 3 X 3 arrays whose entries come from a single row: T(n,k) = det [C(n,k),C(n,k-1),C(n,k-2); C(n,k+1),C(n,k),C(n,k-1); C(n,k+2),C(n,k+1),C(n,k)]. - Peter Bala, May 10 2012

From Gary W. Adamson, Jul 10 2012: (Start)

The triangular view of this triangle is

1;

1, 1;

1, 4, 1;

1, 10, 10, 1;

1, 20, 50, 20, 1;

The n-th row of this triangle is generated by applying the ConvOffs transform to the first n terms of 1, 4, 10, 20, ... (A000292 without leading zero). See A214281 for a procedural definition of the transformation and search "ConvOffs" for more examples. (End)

Define polynomials p(n, x) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -x). If the triangle is extended by the diagonal 1, 0, 0,... on the right side the resulting (0, 0)-based triangle is T*(n, k) = [x^k] p(n, x). The polynomials evaluated at x = 1 gives the number of Baxter permutations of length n (see the formula given by Richard L. Ollerton in A001181). - Peter Luschny, Dec 28 2022

Examples

			The initial rows of the array are:
     1      1      1      1      1      1 ...
     1      4     10     20     35     56 ...
     1     10     50    175    490   1176 ...
     1     20    175    980   4116  14112 ...
     1     35    490   4116  24696 116424 ...
     1     56   1176  14112 116424 731808 ...
     ...
Considered as a triangle, the initial rows are:
  [1],
  [1, 1],
  [1, 4, 1],
  [1, 10, 10, 1],
  [1, 20, 50, 20, 1],
  [1, 35, 175, 175, 35, 1],
  [1, 56, 490, 980, 490, 56, 1],
  [1, 84, 1176, 4116, 4116, 1176, 84, 1],
  [1, 120, 2520, 14112, 24696, 14112, 2520, 120, 1],
  [1, 165, 4950, 41580, 116424, 116424, 41580, 4950, 165, 1],
  [1, 220, 9075, 108900, 457380, 731808, 457380, 108900, 9075, 220, 1]
  ...
		

References

• Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), p. 103-124
• R. P. Stanley, Theory and application of plane partitions. II. Studies in Appl. Math. 50 (1971), p. 259-279. Thm. 18.1

Links

• Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
• J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
• Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
• Johann Cigler, Some observations about Hoggatt triangles, Universität Wien (Austria, 2021).
• Johann Cigler, Some observations about Hankel determinants of the columns of Pascal triangle and related topics, arXiv:2202.07298 [math.CO], 2022.
• Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.
• V. E. Hoggatt, Jr., Letter to N. J. A. Sloane, Apr 1977
• P. A. MacMahon, Combinatory analysis, section 495, 1916.

Crossrefs

Cf. A000372, A056932, A001263, A056940, A056941.

Antidiagonals sum to A001181 (Baxter permutations). Cf. A197208.

Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1..12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

Programs

Maple

# To get initial terms of the array - N. J. A. Sloane, Apr 20 2021
bb := (k,l) -> binomial(k+l,k)*binomial(k+l+1,k)*binomial(k+l+2,k)*2/((k+1)^2*(k+2));
for k from 0 to 8 do
lprint([seq(bb(k,l),l=0..8)]);
od:

Mathematica

t[n_, m_] = 2*Binomial[n, m]*Binomial[n + 1, m + 1]* Binomial[n + 2, m + 2]/((n - m + 1)^2*(n - m + 2)); Flatten[Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]] (* Roger L. Bagula, Jan 28 2009 *)

PARI

\\ cf. A359363
C=binomial;
T(n,k)=if(n==0&&k==0,1,(C(n+1,k-1)*C(n+1,k)*C(n+1,k+1))/(C(n+1,1)*C(n+1,2)));
for(n=1,10,for(k=1,n,print1(T(n,k),", "));print()); \\ Joerg Arndt, Jan 04 2024

Formula

Product_{k=0..2} binomial(n+m+k, m+k)/binomial(n+k, k) gives the array as a square.

T(n,m) = 2*binomial(n, m)*binomial(n+1, m+1)*binomial(n+2, m+2)/((n-m+1)^2*(n-m+2)). - Roger L. Bagula, Jan 28 2009

From Peter Bala, Oct 13 2011: (Start)

T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n+1,k)*C(n+2,k+2)*C(n+3,k+1);

T(n,k) = 2/((n+1)*(n+2)*(n+3))*C(n+1,k+1)*C(n+2,k)*C(n+3,k+2). Cf. A197208.

T(n-1,k-1)*T(n,k+1)*T(n+1,k) = T(n-1,k)*T(n,k-1)*T(n+1,k+1).

Define a(r,n) = n!*(n+1)!*...*(n+r)!. The triangle whose (n,k)-th entry is a(r,0)*a(r,n)/(a(r,k)*a(r,n-k)) is A007318 (r = 0), A001263 (r = 1), A056939 (r = 2), A056940 (r = 3) and A056941 (r = 4). (End)

The column generating functions of the square array (starting at column 1) are 1/(1 - x)^4, (1 + 3*x + x^2)/(1 - x)^7, (1 + 10*x + 20*x^2 + 10*x^3 + x^4)/(1 - x)^10, ..., where the numerator polynomials are the row polynomials of A087647. See Barry p. 31. - Peter Bala, Oct 18 2023