A218577 Triangle read by rows: T(n,k) is the number of ascent sequences of length n with maximal element k-1.

1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 11, 1, 1, 31, 90, 74, 20, 1, 1, 63, 301, 402, 209, 37, 1, 1, 127, 966, 1951, 1629, 590, 70, 1, 1, 255, 3025, 8869, 10839, 6430, 1685, 135, 1, 1, 511, 9330, 38720, 65720, 56878, 25313, 4870, 264, 1

Offset: 1

Author

Joerg Arndt, Nov 03 2012

Keywords

• nonn
• tabl

Comments

Row sums are A022493.

Second column is A000225 (2^n - 1).

Third column appears to be A000392 (Stirling numbers S(n,3)).

Second diagonal (from the right) appears to be A006127 (2^n + n).

Examples

			Triangle starts:
1;
1,    1;
1,    3,     1;
1,    7,     6,      1;
1,   15,    25,     11,      1;
1,   31,    90,     74,     20,      1;
1,   63,   301,    402,    209,     37,      1;
1,  127,   966,   1951,   1629,    590,     70,     1;
1,  255,  3025,   8869,  10839,   6430,   1685,   135,     1;
1,  511,  9330,  38720,  65720,  56878,  25313,  4870,   264,   1;
1, 1023, 28501, 164676, 376114, 444337, 292695, 99996, 14209, 521, 1;
...
The 53 ascent sequences of length 5 are (dots for zeros):
[ #]     ascent-seq.   #max digit
[ 1]    [ . . . . . ]   0
[ 2]    [ . . . . 1 ]   1
[ 3]    [ . . . 1 . ]   1
[ 4]    [ . . . 1 1 ]   1
[ 5]    [ . . . 1 2 ]   2
[ 6]    [ . . 1 . . ]   1
[ 7]    [ . . 1 . 1 ]   1
[ 8]    [ . . 1 . 2 ]   2
[ 9]    [ . . 1 1 . ]   1
[10]    [ . . 1 1 1 ]   1
[11]    [ . . 1 1 2 ]   2
[12]    [ . . 1 2 . ]   2
[13]    [ . . 1 2 1 ]   2
[14]    [ . . 1 2 2 ]   2
[15]    [ . . 1 2 3 ]   3
[16]    [ . 1 . . . ]   1
[17]    [ . 1 . . 1 ]   1
[18]    [ . 1 . . 2 ]   2
[19]    [ . 1 . 1 . ]   1
[20]    [ . 1 . 1 1 ]   1
[21]    [ . 1 . 1 2 ]   2
[22]    [ . 1 . 1 3 ]   3
[23]    [ . 1 . 2 . ]   2
[24]    [ . 1 . 2 1 ]   2
[25]    [ . 1 . 2 2 ]   2
[26]    [ . 1 . 2 3 ]   3
[27]    [ . 1 1 . . ]   1
[28]    [ . 1 1 . 1 ]   1
[29]    [ . 1 1 . 2 ]   2
[...]
[49]    [ . 1 2 3 . ]   3
[50]    [ . 1 2 3 1 ]   3
[51]    [ . 1 2 3 2 ]   3
[52]    [ . 1 2 3 3 ]   3
[53]    [ . 1 2 3 4 ]   4
There is 1 sequence with maximum zero, 15 with maximum one, etc.,
therefore the fifth row is 1, 15, 25, 11, 1.
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened (first 15 rows from Joerg Arndt)
• Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes, Sergey Kitaev, (2+2)-free posets, ascent sequences and pattern avoiding permutations, arXiv:0806.0666 [math.CO], 2008-2009.
• William Y. C. Chen, Alvin Y.L. Dai, Theodore Dokos, Tim Dwyer and Bruce E. Sagan, On 021-Avoiding Ascent Sequences, The Electronic Journal of Combinatorics Volume 20, Issue 1 (2013), #P76.

Crossrefs

Cf. A022493 (number of ascent sequences), A137251 (ascent sequences with k ascents), A175579 (ascent sequences with k zeros).

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).