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A188440
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Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.
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2
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1, 1, 1, 2, 1, 2, 1, 4, 4, 1, 4, 4, 1, 6, 12, 8, 1, 6, 12, 8, 1, 8, 24, 32, 16, 1, 8, 24, 32, 16, 1, 10, 40, 80, 80, 32, 1, 10, 40, 80, 80, 32, 1, 12, 60, 160, 240, 192, 64, 1, 12, 60, 160, 240, 192, 64, 1, 14, 84, 280, 560, 672, 448, 128, 1, 14, 84, 280
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OFFSET
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0,4
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COMMENTS
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A subset S of {1,2,...,n} is antisymmetric if x is an element of S implies n+1-x is not an element of S. In other words, the sum of any two elements of S does not equal n+1. For example, {1,2,5} is an antisymmetric subset of {1,2,3,4,5,6,7}. If n is odd, (n+1)/2 cannot be an element of an antisymmetric subset of {1,2,...,n}. (Note that for n=0, we define {1,...,n} to be the empty set, and thus T(0,0)=1 since the empty set is vacuously antisymmetric.)
We note, for example, that T(100,k) provides the number of possible size-k committees of the U.S. Senate in which no two members are from the same state.
Triangle, with zeros omitted, given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012
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LINKS
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FORMULA
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T(n,k) = 2^k*C(floor(n/2),k) where C(*,*) denotes a binomial coefficient.
Sum(T(n,k),k=0..floor(n/2)) = 3^floor(n/2) = A108411(n).
G.f. for columns(k fixed):(2t^2)^k/((1-t)*(1-t^2)^k).
T(n,k) = T(n-2,k) + 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n.- Philippe Deléham, Apr 09 2012
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EXAMPLE
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Triangle T(n,k) initial values 0 <= k <= floor(n/2), n=0..13:
1
1
1 2
1 2
1 4 4
1 4 4
1 6 12 8
1 6 12 8
1 8 24 32 16
1 8 24 32 16
1 10 40 80 80 32
1 10 40 80 80 32
1 12 60 160 240 192 64
1 12 60 160 240 192 64
...
For n=7 and k=2, T(7,2)=12 since there are 12 antisymmetric size-2 subsets of {1,2,...,7}:
{1,2}, {1,3}, {1,5}, {1,6}, {2,3}, {2,5},
{2,7}, {3,6}, {3,7}, {5,6}, {5,7}, and {6,7}.
(1, 0, -1, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, ...) begins:
1
1 0
1 2 0
1 2 0 0
1 4 4 0 0
1 4 4 0 0 0
1 6 12 8 0 0 0
1 6 12 8 0 0 0 0
1 8 24 32 16 0 0 0 0
1 8 24 32 16 0 0 0 0 0
1 10 40 80 80 32 0 0 0 0 0
1 10 40 80 80 32 0 0 0 0 0 0
1 12 60 160 240 192 64 0 0 0 0 0 0
1 12 60 160 240 192 64 0 0 0 0 0 0 0
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MAPLE
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seq(seq(binomial(floor(n/2), k)*2^k, k=0..floor(n/2)), n=0..22);
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MATHEMATICA
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CROSSREFS
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Cf. A108411, row sums of triangle T(n,k).
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KEYWORD
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nice,easy,nonn,tabf
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AUTHOR
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STATUS
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approved
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