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A218579
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Triangle read by rows: T(n,k) is the number of ascent sequences of length n with last zero at position k-1.
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7
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1, 1, 1, 2, 1, 2, 5, 2, 3, 5, 15, 5, 8, 10, 15, 53, 15, 26, 32, 38, 53, 217, 53, 99, 122, 142, 164, 217, 1014, 217, 433, 537, 619, 704, 797, 1014, 5335, 1014, 2143, 2683, 3069, 3464, 3876, 4321, 5335, 31240, 5335, 11854, 15015, 17063, 19140, 21294, 23522, 25905, 31240
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OFFSET
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1,4
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COMMENTS
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First column and the diagonal is A022493(n-1).
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LINKS
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EXAMPLE
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Triangle starts:
[ 1] 1;
[ 2] 1, 1;
[ 3] 2, 1, 2;
[ 4] 5, 2, 3, 5;
[ 5] 15, 5, 8, 10, 15;
[ 6] 53, 15, 26, 32, 38, 53;
[ 7] 217, 53, 99, 122, 142, 164, 217;
[ 8] 1014, 217, 433, 537, 619, 704, 797, 1014;
[ 9] 5335, 1014, 2143, 2683, 3069, 3464, 3876, 4321, 5335;
[10] 31240, 5335, 11854, 15015, 17063, 19140, 21294, 23522, 25905, 31240;
...
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MAPLE
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b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
add(b(n-1, j, t+`if`(j>i, 1, 0), max(-1, k-1)),
j=`if`(k>=0, 0, 1)..`if`(k=0, 0, t+1)))
end:
T:= (n, k)-> b(n-1, 0, 0, k-2):
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MATHEMATICA
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b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, Sum[b[n-1, j, t + If[j>i, 1, 0], Max[-1, k-1]], {j, If[k >= 0, 0, 1], If[k == 0, 0, t+1]}]]; T[n_, k_] := b[n-1, 0, 0, k-2]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
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CROSSREFS
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Cf. A022493 (number of ascent sequences).
Cf. 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).
Cf. A137251 (ascent sequences with k ascents), A218577 (ascent sequences with maximal element k), A175579 (ascent sequences with k zeros).
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KEYWORD
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AUTHOR
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STATUS
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approved
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