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A175579 Triangle T(n,d) read by rows: Number of ascent sequences of length n with d zeros. 7
1, 1, 1, 2, 2, 1, 5, 6, 3, 1, 15, 21, 12, 4, 1, 53, 84, 54, 20, 5, 1, 217, 380, 270, 110, 30, 6, 1, 1014, 1926, 1490, 660, 195, 42, 7, 1, 5335, 10840, 9020, 4300, 1365, 315, 56, 8, 1, 31240, 67195, 59550, 30290, 10255, 2520, 476, 72, 9, 1, 201608, 455379, 426405 (list; table; graph; refs; listen; history; text; internal format)
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

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-Fran├žois 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).

Sequence in context: A124644 A259691 A056857 * A129100 A162382 A127082

Adjacent sequences:  A175576 A175577 A175578 * A175580 A175581 A175582

KEYWORD

easy,nonn,tabl

AUTHOR

R. J. Mathar, Jul 15 2010

EXTENSIONS

Corrected offset, Joerg Arndt, Nov 03 2012.

STATUS

approved

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Last modified April 27 04:48 EDT 2017. Contains 285507 sequences.