OFFSET
1,4
COMMENTS
The first column and the row sums are both A022493.
Also the number of length-n ascent sequences with k fixed points. [Joerg Arndt, Nov 03 2012]
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened
Hsien-Kuei Hwang, and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.
V. Jelinek, Catalan pairs and Fishburn triples, Adv. Appl. Math. 70 (2015) 1-31
S. Kitaev, J. Remmel, Enumerating (2+2)-free posets by the number of minimal elements and other statistics, Discrete Applied Mathematics 159 (17) (2011), 2098-2108 (preprint: arXiv:1004.3220 [math.CO]).
Paul Levande, Two new interpretations of the Fishburn numbers and their refined generating functions, arXiv:1006.3013
Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, vol.40, pp.945-960 (2001); see p.948.
FORMULA
The bivariate g.f. A(x,y) = Sum_{n>=1, d=1..n} T(n,d)*x^(n+1)*y^(d+1) can be given in two forms (see Remmel and Kitaev, or Levande link):
(1) A(x,y) = Sum_{n>=1} Product_{k=0..n-1} (1 - (1-x)^k*(1-x*y)),
(2) A(x,y) = Sum_{n>=1} x*y/(1-x*y)^n * Product_{k=1..n-1} (1 - (1-x)^k).
EXAMPLE
The triangle starts:
01: 1;
02: 1, 1;
03: 2, 2, 1;
04: 5, 6, 3, 1;
05: 15, 21, 12, 4, 1;
06: 53, 84, 54, 20, 5, 1;
07: 217, 380, 270, 110, 30, 6, 1;
08: 1014, 1926, 1490, 660, 195, 42, 7, 1;
09: 5335, 10840, 9020, 4300, 1365, 315, 56, 8, 1;
10: 31240, 67195, 59550, 30290, 10255, 2520, 476, 72, 9, 1;
11: 201608, 455379, 426405, 229740, 82425, 21448, 4284, 684, 90, 10, 1;
...
From Joerg Arndt, Mar 05 2014: (Start)
The 15 ascent sequences of length 4 (dots for zeros) together with their numbers of zeros and numbers of fixed points are:
01: [ . . . . ] 4 1
02: [ . . . 1 ] 3 1
03: [ . . 1 . ] 3 1
04: [ . . 1 1 ] 2 1
05: [ . . 1 2 ] 2 1
06: [ . 1 . . ] 3 2
07: [ . 1 . 1 ] 2 2
08: [ . 1 . 2 ] 2 2
09: [ . 1 1 . ] 2 2
10: [ . 1 1 1 ] 1 2
11: [ . 1 1 2 ] 1 2
12: [ . 1 2 . ] 2 3
13: [ . 1 2 1 ] 1 3
14: [ . 1 2 2 ] 1 3
15: [ . 1 2 3 ] 1 4
Both statistics give row 4: [5, 6, 3, 1].
(End)
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, 1, expand(add(
`if`(j=0, x, 1) *b(n-1, j, t+`if`(j>i, 1, 0)), j=0..t+1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, -1$2)):
seq(T(n), n=1..12); # Alois P. Heinz, Mar 11 2014
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Expand[Sum[If[j == 0, x, 1]*b[n-1, j, t + If[j>i, 1, 0]], {j, 0, t+1}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, -1, -1]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)
PROG
(PARI) {T(n, d)=polcoeff(polcoeff(sum(m=0, n+1, prod(j=0, m-1, (1-(1-x)^j*(1-x*y) +x^2*y^2*O(x^n*y^d)))), n+1, x), d+1, y)} /* Paul D. Hanna, Feb 18 2012 */
for(n=0, 10, for(d=0, n, print1(T(n, d), ", ")); print(""))
(PARI) {T(n, d)=polcoeff(polcoeff(sum(m=1, n+1, x*y/(1-x*y +x*y*O(x^n*y^d))^m*prod(j=1, m-1, (1-(1-x)^j))), n+1, x), d+1, y)} /* Paul D. Hanna, Feb 18 2012 */
for(n=0, 10, for(d=0, n, print1(T(n, d), ", ")); print(""))
CROSSREFS
Cf. A022493 (number of ascent sequences), A137251 (ascent sequences with k ascents), A218577 (ascent sequences with maximal element k).
Cf. A218579 (ascent sequences with last zero at position k-1), A218580 (ascent sequences with first occurrence of the maximal value at position k-1), A218581 (ascent sequences with last occurrence of the maximal value at position k-1).
T(2n,n) gives A357309.
KEYWORD
AUTHOR
R. J. Mathar, Jul 15 2010
EXTENSIONS
Corrected offset, Joerg Arndt, Nov 03 2012
STATUS
approved