|
| |
|
|
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
|
|
|
REFERENCES
|
M. Rosas, B. Sagan, Symmetric functions in noncommuting variables, arXiv:math.CO/0208168
M. C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. J., 2 (1936), 626-637.
|
|
|
LINKS
|
Table of n, a(n) for n=2..60.
|
|
|
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 (zabrocki(AT)mathstat.yorku.ca), 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}
Comment from Philippe DELEHAM, Jul 16 2007: 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 ;...
|
|
|
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}]] (* From Jean-François Alcover, Oct 04 2011, after given formula *)
|
|
|
CROSSREFS
|
Cf. A008277, A055106, A074664, A000110.
Cf. A055105.
Sequence in context: A203860 A147564 A090981 * A112500 A152938 A154096
Adjacent sequences: A087900 A087901 A087902 * A087904 A087905 A087906
|
|
|
KEYWORD
|
easy,nonn,tabl
|
|
|
AUTHOR
|
Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Oct 14 2003
|
|
|
STATUS
|
approved
|
| |
|
|
