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A218577 Triangle read by rows: T(n,k) is the number of ascent sequences of length n with maximal element k-1. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
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).
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.
EXAMPLE
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.
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).
Sequence in context: A154959 A080417 A008277 * A193387 A185982 A263858
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt, Nov 03 2012
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)