A218579 Triangle read by rows: T(n,k) is the number of ascent sequences of length n with last zero at position k-1.

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

Offset: 1

Author

Joerg Arndt, Nov 03 2012

Keywords

• nonn
• tabl

Comments

Row sums are A022493.

First column and the diagonal is A022493(n-1).

Examples

			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;
...
		

Links

• Alois P. Heinz, Rows n = 1..100, flattened

Crossrefs

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).

Programs

Maple

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):
seq(seq(T(n,k), k=1..n), n=1..10);  # Alois P. Heinz, Nov 16 2012

Mathematica

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 *)