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.

1, 81, 2401, 50625, 923521, 15752961, 260144641, 4228250625, 68184176641, 1095222947841, 17557851463681, 281200199450625, 4501401006735361, 72040003462430721, 1152780773560811521, 18445618199572250625

Offset: 1

Author

Peter C. Heinig (algorithms(AT)gmx.de), Apr 13 2007

Keywords

• easy
• nonn

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}.

Examples

			a(1) = (2^1 - 1)^4 = 1 because only one tuple of length one, namely ({}), has an empty intersection of its sole entry.
		

References

• Stanley, R. P.: Enumerative Combinatorics: Volume 1: Wadsworth & Brooks: 1986: p. 11

Links

• Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024).

Crossrefs

Cf. A000225 (2^n-1), A000583 (n^4).

Programs

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 *)

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]