A369415 Number A(n,k) of n X n Fishburn matrices with entries in the set {0,1,...,k}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 12, 10, 0, 1, 4, 36, 264, 122, 0, 1, 5, 80, 2052, 19632, 3346, 0, 1, 6, 150, 9280, 505764, 4606752, 196082, 0, 1, 7, 252, 30750, 5684480, 511718148, 3311447232, 23869210, 0, 1, 8, 392, 83160, 39378750, 17672135680, 2088275673636, 7202118117504, 5939193962, 0

Offset

0,8

Comments

Number of upper triangular n X n {0,1,...,k}-matrices with no zero rows or columns.

Links

Alois P. Heinz, Antidiagonals n = 0..53, flattened

Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.

Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; arXiv preprint, arXiv:1106.2261 [math.CO], 2011.

Wikipedia, Peter C. Fishburn

Formula

A(n,k) = [x^n] Sum_{j=0..n} x^j * Product_{i=1..j} ((k+1)^i-1)/(1+x*((k+1)^i-1)).

Example

A(2,3) = 3*3*4 = 36:
  [10] [10] [10]  [20] [20] [20]  [30] [30] [30]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
  [11] [11] [11]  [21] [21] [21]  [31] [31] [31]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
  [12] [12] [12]  [22] [22] [22]  [32] [32] [32]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
  [13] [13] [13]  [23] [23] [23]  [33] [33] [33]
  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]  [ 1] [ 2] [ 3]
.
Square array A(n,k) begins:
  1,    1,       1,         1,           1,            1, ...
  0,    1,       2,         3,           4,            5, ...
  0,    2,      12,        36,          80,          150, ...
  0,   10,     264,      2052,        9280,        30750, ...
  0,  122,   19632,    505764,     5684480,     39378750, ...
  0, 3346, 4606752, 511718148, 17672135680, 305416893750, ...
  ...

Maple

A:= (n, k)-> coeff(series(add(x^j*mul(((k+1)^i-1)/(1+x*
           ((k+1)^i-1)), i=1..j), j=0..n), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Crossrefs

Columns k=0-3 give: A000007, A005321, A289314, A289315.

Rows n=0-3 give: A000012, A001477, A011379, A369423.

Main diagonal gives A369336.

Sequence in context: A290605 A292913 A214776 * A317575 A295653 A146326

Adjacent sequences: A369412 A369413 A369414 * A369416 A369417 A369418

Keyword

nonn,tabl

Author

Alois P. Heinz, Jan 22 2024

Status

approved