A133611 A triangular array of numbers related to factorization and number of parts in Murasaki diagrams.

1, 1, 1, 2, 2, 1, 5, 5, 4, 1, 15, 15, 14, 7, 1, 52, 52, 51, 36, 11, 1, 203, 203, 202, 171, 81, 16, 1, 877, 877, 876, 813, 512, 162, 22, 1, 4140, 4140, 4139, 4012, 3046, 1345, 295, 29, 1, 21147, 21147, 21146, 20891, 17866, 10096, 3145, 499, 37, 1, 115975, 115975, 115974, 115463, 106133, 72028, 29503, 6676, 796, 46, 1

Offset: 1

Author

Alford Arnold, Sep 18 2007

Keywords

• nonn
• tabl

Comments

When the Bell multisets are encoded as described in A130274, the seven case in the example can be coded as 19578, 15942, 30873, 26427, 35642, 29491 and 32938.

Examples

			The array begins:
   1
   1  1
   2  2  1
   5  5  4  1
  15 15 14  7  1
  52 52 51 36 11 1
  ...
a(14) = 7 because only seven of the 52 Bell multisets can be generated by attaching a new stroke to the third element in the set of diaqrams with four strokes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10011

Crossrefs

Cf. A000110 (row sums), A137650 (a similar triangle), A130274, A211561.

Cf. A048993.

Programs

Maple

T:= proc(i,j) add(combinat:-stirling2(i,k),k=j..i) end proc:
seq(seq(T(i,j),j=0..i),i=0..15); # Robert Israel, Nov 01 2018
# second Maple program:
b:= proc(n, t) option remember; `if`(n>0, add(b(n-j, t+1)*
      binomial(n-1, j-1), j=1..n), add(x^j, j=0..t))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 30 2019

Mathematica

row[n_] := Table[StirlingS2[n, k], {k, 0, n}] // Reverse // Accumulate // Reverse;
Array[row, 11, 0] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

Formula

Equals A048993 * A000012. - Gary W. Adamson, Jan 29 2008

That is, T(i,j) = Sum_{k=j..i} A048993(i,k) for 0 <= j <= i. - Robert Israel, Nov 01 2018

Extensions

Definition not clear to me - N. J. A. Sloane, Sep 18 2007

More terms from Robert Israel, Nov 01 2018