A137251 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.

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, 1, 1, 28, 322, 1433, 2285, 1138, 127, 1, 1, 36, 588, 4066, 11210, 11413, 3670, 255, 1, 1, 45, 1002, 10165, 44443, 79781, 54073, 11586, 511, 1, 1, 55, 1617, 23056, 150546, 434638, 528690, 246409, 36038, 1023, 1

Offset: 1

Author

Vladeta Jovovic, Mar 11 2008

Keywords

• nonn
• tabl

Comments

Row sums are A022493.

Number of ascent sequences of length n with k-1 ascents, see example. [Joerg Arndt, Nov 03 2012]

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: yVít Jelínek, Sep 06 2014

Examples

			Triangle starts:
01:  1,
02:  1,  1,
03:  1,  3,    1,
04:  1,  6,    7,     1,
05:  1, 10,   26,    15,      1,
06:  1, 15,   71,    98,     31,       1,
07:  1, 21,  161,   425,    342,      63,       1,
08:  1, 28,  322,  1433,   2285,    1138,     127,       1,
09:  1, 36,  588,  4066,  11210,   11413,    3670,     255,       1,
10:  1, 45, 1002, 10165,  44443,   79781,   54073,   11586,     511,      1,
11:  1, 55, 1617, 23056, 150546,  434638,  528690,  246409,   36038,   1023,    1,
12:  1, 66, 2497, 48400, 451515, 1968580, 3895756, 3316193, 1090517, 110930, 2047, 1,
...
From _Joerg Arndt_, Nov 03 2012: (Start)
The 53 ascent sequences of length 5 together with their numbers of ascents are (dots for zeros):
01:  [ . . . . . ]   0      28:  [ . 1 1 . 1 ]   2
02:  [ . . . . 1 ]   1      29:  [ . 1 1 . 2 ]   2
03:  [ . . . 1 . ]   1      30:  [ . 1 1 1 . ]   1
04:  [ . . . 1 1 ]   1      31:  [ . 1 1 1 1 ]   1
05:  [ . . . 1 2 ]   2      32:  [ . 1 1 1 2 ]   2
06:  [ . . 1 . . ]   1      33:  [ . 1 1 2 . ]   2
07:  [ . . 1 . 1 ]   2      34:  [ . 1 1 2 1 ]   2
08:  [ . . 1 . 2 ]   2      35:  [ . 1 1 2 2 ]   2
09:  [ . . 1 1 . ]   1      36:  [ . 1 1 2 3 ]   3
10:  [ . . 1 1 1 ]   1      37:  [ . 1 2 . . ]   2
11:  [ . . 1 1 2 ]   2      38:  [ . 1 2 . 1 ]   3
12:  [ . . 1 2 . ]   2      39:  [ . 1 2 . 2 ]   3
13:  [ . . 1 2 1 ]   2      40:  [ . 1 2 . 3 ]   3
14:  [ . . 1 2 2 ]   2      41:  [ . 1 2 1 . ]   2
15:  [ . . 1 2 3 ]   3      42:  [ . 1 2 1 1 ]   2
16:  [ . 1 . . . ]   1      43:  [ . 1 2 1 2 ]   3
17:  [ . 1 . . 1 ]   2      44:  [ . 1 2 1 3 ]   3
18:  [ . 1 . . 2 ]   2      45:  [ . 1 2 2 . ]   2
19:  [ . 1 . 1 . ]   2      46:  [ . 1 2 2 1 ]   2
20:  [ . 1 . 1 1 ]   2      47:  [ . 1 2 2 2 ]   2
21:  [ . 1 . 1 2 ]   3      48:  [ . 1 2 2 3 ]   3
22:  [ . 1 . 1 3 ]   3      49:  [ . 1 2 3 . ]   3
23:  [ . 1 . 2 . ]   2      50:  [ . 1 2 3 1 ]   3
24:  [ . 1 . 2 1 ]   2      51:  [ . 1 2 3 2 ]   3
25:  [ . 1 . 2 2 ]   2      52:  [ . 1 2 3 3 ]   3
26:  [ . 1 . 2 3 ]   3      53:  [ . 1 2 3 4 ]   4
27:  [ . 1 1 . . ]   1
There is 1 ascent sequence with no ascent, 10 with one ascent, etc., giving the fourth row [1, 10, 26, 15, 1].
(End)
		

References

• Peter C. Fishburn, Interval Orders and Interval Graphs: Study of Partially Ordered Sets, John Wiley & Sons, 1985.

Links

• Alois P. Heinz, Rows n = 1..141, flattened (Rows n = 1..15 from Joerg Arndt)
• Mireille Bousquet-Mélou, Anders Claesson, Mark Dukes, and Sergey Kitaev, (2+2)-free posets, ascent sequences and pattern avoiding permutations, arXiv:0806.0666 [math.CO], 2008.
• Giulio Cerbai, Modified ascent sequences and Bell numbers, arXiv:2305.10820 [math.CO], 2023. See p. 8.
• 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.
• Matthieu Dien, Antoine Genitrini, and Frederic Peschanski, A Combinatorial Study of Async/Await Processes, Conf.: 19th Int'l Colloq. Theor. Aspects of Comp. (2022), (Analytic) Combinatorics of concurrent systems.
• M. Dukes, V Jelínek, and M. Kubitzke Composition Matrices, (2+2)-Free Posets and their Specializations, Electronic Journal of Combinatorics, Volume 18, Issue 1, 2011, Paper #P44.
• Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614; arXiv preprint, arXiv:1106.2261 [math.CO], 2011.

Crossrefs

Cf. A022493 (number of ascent sequences), A218577 (ascent sequences with maximal element k), A175579 (ascent sequences with k zeros).

T(2n,n) gives A357141.

Programs

Maple

b:= proc(n, i, t) option remember; local j; if n<1 then [0$t, 1]
      else []; for j from 0 to t+1 do zip((x, y)->x+y, %,
      b(n-1, j, t+`if`(j>i, 1, 0)), 0) od; % fi
    end:
T:= n-> b(n-1, 0, 0)[]:
seq(T(n), n=1..12);  # Alois P. Heinz, May 20 2013

Mathematica

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 *)

Formula

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