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A175579
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Triangle T(n,d) read by rows: Number of ascent sequences of length n with d zeros.
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7
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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
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OFFSET
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1,4
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COMMENTS
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The first column and the row sums are both A022493.
T(n,k) also count the number of length-n ascent sequences with k fixed points. [Joerg Arndt, Nov 03 2012]
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LINKS
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Joerg Arndt, Table of n, a(n) for n = 1..1225
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
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FORMULA
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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).
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EXAMPLE
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The triangle starts:
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;
201608,455379,426405,229740,82425,21448,4284,684,90,10,1;
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PROG
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(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(""))
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CROSSREFS
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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: A171670 A124644 A056857 * A129100 A162382 A127082
Adjacent sequences: A175576 A175577 A175578 * A175580 A175581 A175582
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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R. J. Mathar, Jul 15 2010
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EXTENSIONS
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Corrected offset, Joerg Arndt, Nov 03 2012.
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STATUS
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approved
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