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%I #33 Nov 26 2018 17:12:49
%S 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,
%T 1,1,127,966,1951,1629,590,70,1,1,255,3025,8869,10839,6430,1685,135,1,
%U 1,511,9330,38720,65720,56878,25313,4870,264,1
%N Triangle read by rows: T(n,k) is the number of ascent sequences of length n with maximal element k-1.
%C Row sums are A022493.
%C Second column is A000225 (2^n - 1).
%C Third column appears to be A000392 (Stirling numbers S(n,3)).
%C Second diagonal (from the right) appears to be A006127 (2^n + n).
%H Joerg Arndt and Alois P. Heinz, <a href="/A218577/b218577.txt">Rows n = 1..141, flattened</a> (first 15 rows from Joerg Arndt)
%H Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes, Sergey Kitaev, <a href="http://arxiv.org/abs/0806.0666">(2+2)-free posets, ascent sequences and pattern avoiding permutations</a>, arXiv:0806.0666 [math.CO], 2008-2009.
%H William Y. C. Chen, Alvin Y.L. Dai, Theodore Dokos, Tim Dwyer and Bruce E. Sagan, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p76">On 021-Avoiding Ascent Sequences, The Electronic Journal of Combinatorics</a> Volume 20, Issue 1 (2013), #P76.
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 7, 6, 1;
%e 1, 15, 25, 11, 1;
%e 1, 31, 90, 74, 20, 1;
%e 1, 63, 301, 402, 209, 37, 1;
%e 1, 127, 966, 1951, 1629, 590, 70, 1;
%e 1, 255, 3025, 8869, 10839, 6430, 1685, 135, 1;
%e 1, 511, 9330, 38720, 65720, 56878, 25313, 4870, 264, 1;
%e 1, 1023, 28501, 164676, 376114, 444337, 292695, 99996, 14209, 521, 1;
%e ...
%e The 53 ascent sequences of length 5 are (dots for zeros):
%e [ #] ascent-seq. #max digit
%e [ 1] [ . . . . . ] 0
%e [ 2] [ . . . . 1 ] 1
%e [ 3] [ . . . 1 . ] 1
%e [ 4] [ . . . 1 1 ] 1
%e [ 5] [ . . . 1 2 ] 2
%e [ 6] [ . . 1 . . ] 1
%e [ 7] [ . . 1 . 1 ] 1
%e [ 8] [ . . 1 . 2 ] 2
%e [ 9] [ . . 1 1 . ] 1
%e [10] [ . . 1 1 1 ] 1
%e [11] [ . . 1 1 2 ] 2
%e [12] [ . . 1 2 . ] 2
%e [13] [ . . 1 2 1 ] 2
%e [14] [ . . 1 2 2 ] 2
%e [15] [ . . 1 2 3 ] 3
%e [16] [ . 1 . . . ] 1
%e [17] [ . 1 . . 1 ] 1
%e [18] [ . 1 . . 2 ] 2
%e [19] [ . 1 . 1 . ] 1
%e [20] [ . 1 . 1 1 ] 1
%e [21] [ . 1 . 1 2 ] 2
%e [22] [ . 1 . 1 3 ] 3
%e [23] [ . 1 . 2 . ] 2
%e [24] [ . 1 . 2 1 ] 2
%e [25] [ . 1 . 2 2 ] 2
%e [26] [ . 1 . 2 3 ] 3
%e [27] [ . 1 1 . . ] 1
%e [28] [ . 1 1 . 1 ] 1
%e [29] [ . 1 1 . 2 ] 2
%e [...]
%e [49] [ . 1 2 3 . ] 3
%e [50] [ . 1 2 3 1 ] 3
%e [51] [ . 1 2 3 2 ] 3
%e [52] [ . 1 2 3 3 ] 3
%e [53] [ . 1 2 3 4 ] 4
%e There is 1 sequence with maximum zero, 15 with maximum one, etc.,
%e therefore the fifth row is 1, 15, 25, 11, 1.
%Y Cf. A022493 (number of ascent sequences), A137251 (ascent sequences with k ascents), A175579 (ascent sequences with k zeros).
%Y 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).
%K nonn,tabl
%O 1,5
%A _Joerg Arndt_, Nov 03 2012
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