|
| |
|
|
A128833
|
|
Number of n-tuples where each entry is chosen from the subsets of {1,2,3,4,5} such that the intersection of all n entries is empty.
|
|
2
|
|
|
|
1, 243, 16807, 759375, 28629151, 992436543, 33038369407, 1078203909375, 34842114263551, 1120413075641343, 35940921946155007, 1151514816750309375, 36870975646169341951, 1180231376725002502143, 37773167607267111108607
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
The general formula where each entry is chosen from the subsets of {1,..,k} is (2^n-1)^k. This may be shown by exhibiting a bijection to a set whose cardinality is obviously (2^n-1)^k, namely the set of all k-tuples with each entry chosen from the 2^n-1 proper subsets of {1,..,n}, i.e. for of the k entries {1,..,n} is forbidden. The bijection is given by (X_1,..,X_n) |-> (Y_1,..,Y_k) where for each j in {1,..,k} and each i in {1,..,n}, i is in Y_j if and only if j is in X_i. Sequence A060867 is the case where the entries are chosen from subsets of {1,2}.
|
|
|
REFERENCES
|
Richard P. Stanley, Enumerative Combinatorics, Volume 1, Wadsworth & Brooks, 1986, p. 11.
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) = (2^n-1)^5
G.f.: x*(1024*x^4+5760*x^3+2800*x^2+180*x+1)/((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)*(32*x-1)). [Colin Barker, Nov 17 2012]
|
|
|
EXAMPLE
|
a(1)=(2^1-1)^5=1 because only one tuple of length one, namely ({}) has an empty intersection of its sole entry.
|
|
|
MAPLE
|
for k from 1 to 20 do (2^k-1)^5; od;
|
|
|
MATHEMATICA
|
(2^Range[20]-1)^5 (* or *) LinearRecurrence[{63, -1302, 11160, -41664, 64512, -32768}, {1, 243, 16807, 759375, 28629151, 992436543}, 20] (* or *) CoefficientList[Series[x (1024x^4+5760x^3+2800x^2+180x+1)/((x-1)(2x-1)(4x-1)(8x-1)(16x-1)(32x-1)), {x, 0, 20}], x] (* Harvey P. Dale, Aug 16 2021 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
easy,nonn
|
|
|
AUTHOR
|
Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007
|
|
|
STATUS
|
approved
|
| |
|
|
