A125311 Array giving number of (k,2)-noncrossing partitions of [n], read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 5, 14, 1, 1, 2, 5, 15, 42, 1, 1, 2, 5, 15, 51, 132, 1, 1, 2, 5, 15, 52, 188, 429, 1, 1, 2, 5, 15, 52, 202, 731, 1430, 1, 1, 2, 5, 15, 52, 203, 856, 2950, 4862, 1, 1, 2, 5, 15, 52, 203, 876, 3868, 12235, 16796

Offset

0,6

Comments

A partition is (k,2)-noncrossing if it avoids the pattern 12...k12.

Links

Table of n, a(n) for n=0..65.

Toufik Mansour and Simone Severini, Enumeration of (k,2)-noncrossing partitions, Discrete Math., 308 (2008), 4570-4577.

Example

Table begins:
k\n|  0   1   2   3   4   5     6     7     8     9      10     11      12
  2|  1   1   2   5  14  42   132   429  1430  4862   16796  58786  208012
  3|  1   1   2   5  15  51   188   731  2950 12235   51822 223191  974427
  4|  1   1   2   5  15  52   202   856  3868 18313   89711 450825 2310453
  5|  1   1   2   5  15  52   203   876  4112 20679  109853 608996 3488806
  6|  1   1   2   5  15  52   203   877  4139 21111  115219 666388 4045991

Mathematica

b[j_, j_] := 1;
b[i_, j_] := j x Product[s x - 1, {s, i + 1, j - 1}];
y[k_] := (1 - (k - 2) x - Sqrt[(1 - k x)^2 - 4 x^2]) / (2 x (1 - (k - 2) x));
s[k_, op_] := Sum[(-1)^(i + j) op[x, i] b[i, j], {j, 0, k - 2}, {i, 0, j}];
p[k_] := (x^(k - 1) y[k]/(1 - x y[k]) + s[k, Power]) / (1 - s[k, Times]);
t[n_, k_] := SeriesCoefficient[p[k], {x, 0, n}];
Print@Flatten@Table[t[n, ad - n + 2], {ad, 0, 10}, {n, 0, ad}]
(* Andrey Zabolotskiy, Sep 14 2021 *)

Crossrefs

Rows include A000108, A007317, A140980, A141080, A141081.

Cf. A000110.

Sequence in context: A000361 A246596 A135723 * A341359 A127568 A263791

Adjacent sequences: A125308 A125309 A125310 * A125312 A125313 A125314

Keyword

nonn,tabl

Author

Jonathan Vos Post, Dec 10 2006

Extensions

Offset corrected by Joerg Arndt, Apr 18 2014

More terms from Andrey Zabolotskiy, Sep 14 2021

Status

approved