A087903 Triangle read by rows of the numbers T(n,k) (n > 1, 0 < k < n) of set partitions of n of length k which do not have a proper subset of parts with a union equal to a subset {1,2,...,j} with j < n.

1, 1, 1, 1, 4, 1, 1, 11, 9, 1, 1, 26, 48, 16, 1, 1, 57, 202, 140, 25, 1, 1, 120, 747, 916, 325, 36, 1, 1, 247, 2559, 5071, 3045, 651, 49, 1, 1, 502, 8362, 25300, 23480, 8260, 1176, 64, 1, 1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1, 1, 2036, 82509, 525608, 998830, 749154, 253764, 40944, 3105, 100, 1

Offset: 2

Author

Mike Zabrocki, Oct 14 2003

Keywords

• easy
• nonn
• tabl

Comments

Another version of the triangle T(n,k), 0 <= k <= n, given by [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938; see also A086329 for a triangle transposed. - Philippe Deléham, Jun 13 2004

Examples

			T(2,1)=1 for {12};
T(3,1)=1, T(3,2) = 1 for {123}; {13|2};
T(4,1)=1, T(4,2)=4, T(4,3)=1 for {1234}; {14|23}, {13|24}, {124|3}, {134|2}; {14|2|3}.
From _Philippe Deléham_, Jul 16 2007: (Start)
Triangle begins:
  1;
  1,    1;
  1,    4,     1;
  1,   11,     9,      1;
  1,   26,    48,     16,      1;
  1,   57,   202,    140,     25,     1;
  1,  120,   747,    916,    325,    36,     1;
  1,  247,  2559,   5071,   3045,   651,    49,    1;
  1,  502,  8362,  25300,  23480,  8260,  1176,   64,  1;
  1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1;
  ...
Triangle T(n,k), 0 <= k <= n, given by [1,0,2,0,3,0,4,0,...] DELTA [0,1,0,1,0,1,0,...] begins:
  1;
  1,    0;
  1,    1,     0;
  1,    4,     1,      0;
  1,   11,     9,      1,      0;
  1,   26,    48,     16,      1,     0;
  1,   57,   202,    140,     25,     1,     0;
  1,  120,   747,    916,    325,    36,     1,    0;
  1,  247,  2559,   5071,   3045,   651,    49,    1,  0;
  1,  502,  8362,  25300,  23480,  8260,  1176,   64,  1, 0;
  1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1, 0;
  ...
(End)
		

Links

• G. C. Greubel, Rows n = 2..52 of the triangle, flattened
• V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
• Mercedes H. Rosas and Bruce E. Sagan, Symmetric functions in noncommuting variables, arXiv:math/0208168 [math.CO], 2002, 2004.
• Yves Le Jan, Loop clusters on complete graphs, arXiv:2504.16976 [math.PR], 2025. See p. 6.
• Margarete C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. J., 2 (1936), 626-637.

Crossrefs

Cf. A000110, A000295, A008277, A055106, A074664.

Cf. A055105.

Programs

Maple

A := proc(n,k) option remember; local j,ell; if n<=0 or k>=n then 0; elif k=1 or k=n-1 then 1; else S2(n-1,k)+add(add((k-ell-1)*A(n-j-1,k-ell)*S2(j,ell),ell=0..k-1),j=0..n-2); fi; end: S2 := (n,k)->if n<0 or k>n then 0; elif k=n or k=1 then 1 else k*S2(n-1,k)+S2(n-1,k-1); fi:

Mathematica

nmax = 12; t[n_, k_] := t[n, k] = StirlingS2[n-1, k] + Sum[ (k-d-1)*t[n-j-1, k-d]*StirlingS2[j, d], {d, 0, k-1}, {j, 0, n-2}]; Flatten[ Table[ t[n, k], {n, 2, nmax}, {k, 1, n-1}]] (* Jean-François Alcover, Oct 04 2011, after given formula *)

SageMath

@CachedFunction # T = A087903
def T(n,k): return stirling_number2(n-1, k) + sum( sum( (k-m-1)*T(n-j-1, k-m)*stirling_number2(j, m) for m in (0..k-1) ) for j in (0..n-2) )
flatten([[T(n, k) for k in (1..n-1)] for n in (2..14)]) # G. C. Greubel, Jun 21 2022

Formula

T(n, n-1) = T(n,1) = 1.

T(n, n-2) = (n-2)^2.

T(n, 2) = A000295(n).

T(n, k) = S2(n-1, k) + Sum_{j=0..n-2} Sum_{d=0..k-1} (k-d-1)*T(n-j-1, k-d)*S2(j, d), where S2(n, k) is the Stirling number of the second kind.

Sum_{k = 1..n-1} T(n, k) = A074664(n). - Philippe Deléham, Jun 13 2004

G.f.: 1-1/(1+add(add(q^n t^k S2(n, k), k=1..n), n >= 1)) where S2(n, k) are the Stirling numbers of the 2nd kind A008277. - Mike Zabrocki, Sep 03 2005