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%I #59 May 24 2023 13:15:06
%S 1,1,1,1,3,1,1,6,7,1,1,10,26,15,1,1,15,71,98,31,1,1,21,161,425,342,63,
%T 1,1,28,322,1433,2285,1138,127,1,1,36,588,4066,11210,11413,3670,255,1,
%U 1,45,1002,10165,44443,79781,54073,11586,511,1,1,55,1617,23056,150546,434638,528690,246409,36038,1023,1
%N Triangle T(n,k) read by rows: number of k X k triangular matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n, n>=1, 1<=k<=n.
%C Row sums are A022493.
%C Number of ascent sequences of length n with k-1 ascents, see example. [_Joerg Arndt_, Nov 03 2012]
%C Number of interval orders on n elements having exactly k maximal antichains. Also, number of interval orders on n elements having an interval representation with k distinct endpoints, but not with k-1 distinct endpoints. Also, number of interval orders on n elements whose elements define k distinct strict down-sets (a strict down-set defined by an element x of a poset (P,<) is the set {y in P: y<x}). See Fishburn, Chapter 2.3. - _Vít Jelínek_, Sep 06 2014
%D Peter C. Fishburn, Interval Orders and Interval Graphs: Study of Partially Ordered Sets, John Wiley & Sons, 1985.
%H Alois P. Heinz, <a href="/A137251/b137251.txt">Rows n = 1..141, flattened</a> (Rows n = 1..15 from Joerg Arndt)
%H Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes, and 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.
%H Giulio Cerbai, <a href="https://arxiv.org/abs/2305.10820">Modified ascent sequences and Bell numbers</a>, arXiv:2305.10820 [math.CO], 2023. See p. 8.
%H William Y. C. Chen, Alvin Y.L. Dai, Theodore Dokos, Tim Dwyer and Bruce E. Sagan, <a href="https://doi.org/10.37236/2472">On 021-Avoiding Ascent Sequences, The Electronic Journal of Combinatorics</a> Volume 20, Issue 1 (2013), #P76.
%H Matthieu Dien, Antoine Genitrini, and Frederic Peschanski, <a href="https://www.researchgate.net/publication/363253998_A_Combinatorial_Study_of_AsyncAwait_Processes">A Combinatorial Study of Async/Await Processes</a>, Conf.: 19th Int'l Colloq. Theor. Aspects of Comp. (2022), (Analytic) Combinatorics of concurrent systems.
%H M. Dukes, V Jelínek, and M. Kubitzke <a href="https://doi.org/10.37236/531"> Composition Matrices, (2+2)-Free Posets and their Specializations</a>, Electronic Journal of Combinatorics, Volume 18, Issue 1, 2011, Paper #P44.
%H Vít Jelínek, <a href="http://dx.doi.org/10.1016/j.jcta.2011.11.010">Counting general and self-dual interval orders</a>, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; <a href="http://arxiv.org/abs/1106.2261">arXiv preprint</a>, arXiv:1106.2261 [math.CO], 2011.
%F G.f.: Sum_{n,k>=1} T(n,k)x^n y^k = Sum_{n>=1} y^n Product_{i=1..n} (1-(1-x)^i)/(y+(1-x)^i-y*(1-x)^i). See Jelínek's paper, Corollary 2.5. - _Vít Jelínek_, Sep 06 2014
%e Triangle starts:
%e 01: 1,
%e 02: 1, 1,
%e 03: 1, 3, 1,
%e 04: 1, 6, 7, 1,
%e 05: 1, 10, 26, 15, 1,
%e 06: 1, 15, 71, 98, 31, 1,
%e 07: 1, 21, 161, 425, 342, 63, 1,
%e 08: 1, 28, 322, 1433, 2285, 1138, 127, 1,
%e 09: 1, 36, 588, 4066, 11210, 11413, 3670, 255, 1,
%e 10: 1, 45, 1002, 10165, 44443, 79781, 54073, 11586, 511, 1,
%e 11: 1, 55, 1617, 23056, 150546, 434638, 528690, 246409, 36038, 1023, 1,
%e 12: 1, 66, 2497, 48400, 451515, 1968580, 3895756, 3316193, 1090517, 110930, 2047, 1,
%e ...
%e From _Joerg Arndt_, Nov 03 2012: (Start)
%e The 53 ascent sequences of length 5 together with their numbers of ascents are (dots for zeros):
%e 01: [ . . . . . ] 0 28: [ . 1 1 . 1 ] 2
%e 02: [ . . . . 1 ] 1 29: [ . 1 1 . 2 ] 2
%e 03: [ . . . 1 . ] 1 30: [ . 1 1 1 . ] 1
%e 04: [ . . . 1 1 ] 1 31: [ . 1 1 1 1 ] 1
%e 05: [ . . . 1 2 ] 2 32: [ . 1 1 1 2 ] 2
%e 06: [ . . 1 . . ] 1 33: [ . 1 1 2 . ] 2
%e 07: [ . . 1 . 1 ] 2 34: [ . 1 1 2 1 ] 2
%e 08: [ . . 1 . 2 ] 2 35: [ . 1 1 2 2 ] 2
%e 09: [ . . 1 1 . ] 1 36: [ . 1 1 2 3 ] 3
%e 10: [ . . 1 1 1 ] 1 37: [ . 1 2 . . ] 2
%e 11: [ . . 1 1 2 ] 2 38: [ . 1 2 . 1 ] 3
%e 12: [ . . 1 2 . ] 2 39: [ . 1 2 . 2 ] 3
%e 13: [ . . 1 2 1 ] 2 40: [ . 1 2 . 3 ] 3
%e 14: [ . . 1 2 2 ] 2 41: [ . 1 2 1 . ] 2
%e 15: [ . . 1 2 3 ] 3 42: [ . 1 2 1 1 ] 2
%e 16: [ . 1 . . . ] 1 43: [ . 1 2 1 2 ] 3
%e 17: [ . 1 . . 1 ] 2 44: [ . 1 2 1 3 ] 3
%e 18: [ . 1 . . 2 ] 2 45: [ . 1 2 2 . ] 2
%e 19: [ . 1 . 1 . ] 2 46: [ . 1 2 2 1 ] 2
%e 20: [ . 1 . 1 1 ] 2 47: [ . 1 2 2 2 ] 2
%e 21: [ . 1 . 1 2 ] 3 48: [ . 1 2 2 3 ] 3
%e 22: [ . 1 . 1 3 ] 3 49: [ . 1 2 3 . ] 3
%e 23: [ . 1 . 2 . ] 2 50: [ . 1 2 3 1 ] 3
%e 24: [ . 1 . 2 1 ] 2 51: [ . 1 2 3 2 ] 3
%e 25: [ . 1 . 2 2 ] 2 52: [ . 1 2 3 3 ] 3
%e 26: [ . 1 . 2 3 ] 3 53: [ . 1 2 3 4 ] 4
%e 27: [ . 1 1 . . ] 1
%e There is 1 ascent sequence with no ascent, 10 with one ascent, etc., giving the fourth row [1, 10, 26, 15, 1].
%e (End)
%p b:= proc(n, i, t) option remember; local j; if n<1 then [0$t, 1]
%p else []; for j from 0 to t+1 do zip((x, y)->x+y, %,
%p b(n-1, j, t+`if`(j>i, 1, 0)), 0) od; % fi
%p end:
%p T:= n-> b(n-1, 0, 0)[]:
%p seq(T(n), n=1..12); # _Alois P. Heinz_, May 20 2013
%t zip[f_, x_List, y_List, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]]; b[n_, i_, t_] := b[n, i, t] = Module[{j, pc}, If[n<1, Append[Array[0&, t], 1], pc = {}; For[j = 0, j <= t+1, j++, pc = zip[Plus, pc, b[n-1, j, t+If[j>i, 1, 0]], 0]]; pc]]; T[n_] := b[n-1, 0, 0]; Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Jan 29 2014, after _Alois P. Heinz_ *)
%Y Cf. A022493 (number of ascent sequences), A218577 (ascent sequences with maximal element k), A175579 (ascent sequences with k zeros).
%Y T(2n,n) gives A357141.
%K nonn,tabl
%O 1,5
%A _Vladeta Jovovic_, Mar 11 2008
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