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Search: id:A128833
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| A128833 |
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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. |
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2
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| 1, 243, 16807, 759375, 28629151, 992436543, 33038369407, 1078203909375, 34842114263551, 1120413075641343, 35940921946155007, 1151514816750309375, 36870975646169341951, 1180231376725002502143, 37773167607267111108607
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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}.
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REFERENCES
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Stanley, R.P.: Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11
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FORMULA
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a(n)=(2^n-1)^5
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EXAMPLE
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a(1)=(2^1-1)^5=1 because only one tuple of length one, namely ({}) has an
empty intersection of its sole entry.
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MAPLE
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for k from 1 to 20 do (2^k-1)^5; od;
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CROSSREFS
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Cf. A060867.
Sequence in context: A016769 A059860 A016841 this_sequence A016889 A016949 A086649
Adjacent sequences: A128830 A128831 A128832 this_sequence A128834 A128835 A128836
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KEYWORD
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easy,nonn
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AUTHOR
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Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007
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