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 A128832 Number of n-tuples where each entry is chosen from the subsets of {1,2,3,4} such that the intersection of all n entries is empty. 2
 1, 81, 2401, 50625, 923521, 15752961, 260144641, 4228250625, 68184176641, 1095222947841, 17557851463681, 281200199450625, 4501401006735361, 72040003462430721, 1152780773560811521, 18445618199572250625 (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 Stanley, R. P.: Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11 LINKS Table of n, a(n) for n=1..16. Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024). FORMULA a(n) = (2^n - 1)^4. G.f.: -x*(4*x+1)*(16*x^2+46*x+1)/((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)). [Colin Barker, Nov 17 2012] EXAMPLE a(1) = (2^1 - 1)^4 = 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)^4; od; with (combinat):seq(mul(stirling2(n, 2), k=1..4), n=2..17); # Zerinvary Lajos, Dec 16 2007 MATHEMATICA LinearRecurrence[{31, -310, 1240, -1984, 1024}, {1, 81, 2401, 50625, 923521}, 20] (* Harvey P. Dale, Mar 30 2019 *) CROSSREFS Cf. A000225 (2^n-1), A000583 (n^4). Sequence in context: A236989 A038676 A016840 * A085877 A123219 A205512 Adjacent sequences: A128829 A128830 A128831 * A128833 A128834 A128835 KEYWORD easy,nonn AUTHOR Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007 STATUS approved

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Last modified November 29 03:03 EST 2023. Contains 367422 sequences. (Running on oeis4.)