A070925 - OEIS

A070925

Number of subsets of A = {1,2,...,n} that have the same center of gravity as A, i.e., (n+1)/2.

1, 1, 3, 3, 7, 7, 19, 17, 51, 47, 151, 137, 471, 427, 1519, 1391, 5043, 4651, 17111, 15883, 59007, 55123, 206259, 193723, 729095, 688007, 2601639, 2465133, 9358943, 8899699, 33904323, 32342235, 123580883, 118215779, 452902071, 434314137, 1667837679, 1602935103

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OFFSET

1,3

COMMENTS

From Gus Wiseman, Apr 15 2023: (Start)

Also the number of nonempty subsets of {0..n} with mean n/2. The a(0) = 1 through a(5) = 7 subsets are:

{0} {0,1} {1} {0,3} {2} {0,5}

{0,2} {1,2} {0,4} {1,4}

{0,1,2} {0,1,2,3} {1,3} {2,3}

{0,2,4} {0,1,4,5}

{1,2,3} {0,2,3,5}

{0,1,3,4} {1,2,3,4}

{0,1,2,3,4} {0,1,2,3,4,5}

(End)

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..40

FORMULA

From Gus Wiseman, Apr 18 2023: (Start)

a(2n+1) = A000980(n) - 1.

a(n) = A222955(n) - 1.

a(n) = 2*A362046(n) + 1.

(End)

EXAMPLE

Of the 32 (2^5) sets which can be constructed from the set A = {1,2,3,4,5} only the sets {3}, {2, 3, 4}, {2, 4}, {1, 2, 4, 5}, {1, 2, 3, 4, 5}, {1, 3, 5}, {1, 5} give an average of 3.

MATHEMATICA

Needs["DiscreteMath`Combinatorica`"]; f[n_] := Block[{s = Subsets[n], c = 0, k = 2}, While[k < 2^n + 1, If[ (Plus @@ s[[k]]) / Length[s[[k]]] == (n + 1)/2, c++ ]; k++ ]; c]; Table[ f[n], {n, 1, 20}]

(* second program *)

Table[Length[Select[Subsets[Range[0, n]], Mean[#]==n/2&]], {n, 0, 10}] (* Gus Wiseman, Apr 15 2023 *)

CROSSREFS

The odd bisection is A000980(n) - 1 = 2*A047653(n) - 1.

For median instead of mean we have A100066, bisection A006134.

Including the empty set gives A222955.

The one-based version is A362046, even bisection A047653(n) - 1.

A007318 counts subsets by length.

A067538 counts partitions with integer mean, strict A102627.

A231147 counts subsets by median.

A327481 counts subsets by integer mean.

Cf. A024718, A057552, A079309, A133406, A326512, A326513, A326537, A327475, A349156, A359893.