login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 11
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,5

COMMENTS

T(n,n-1) = T(n,1) = 1; T(n,n-2) = (n-2)^2; T(n,2) = A000295(n).

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

LINKS

Table of n, a(n) for n=2..60.

V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.

M. Rosas, B. Sagan, Symmetric functions in noncommuting variables, arXiv:math/0208168 [math.CO], 2002, 2004.

M. C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. J., 2 (1936), 626-637.

FORMULA

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

EXAMPLE

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)

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 *)

CROSSREFS

Cf. A008277, A055106, A074664, A000110.

Cf. A055105.

Sequence in context: A203860 A147564 A090981 * A287532 A112500 A152938

Adjacent sequences:  A087900 A087901 A087902 * A087904 A087905 A087906

KEYWORD

easy,nonn,tabl

AUTHOR

Mike Zabrocki, Oct 14 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 22 22:19 EDT 2021. Contains 343197 sequences. (Running on oeis4.)