A188440 Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.

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

Offset: 0

Author

Dennis P. Walsh, Mar 31 2011

Keywords

• nice
• easy
• nonn
• tabf

Comments

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, read by rows, A013609 rows repeated. - Philippe Deléham, Apr 09 2012

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

Examples

			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
- _Philippe Deléham_, Apr 09 2012
		

Links

• T. D. Noe, Rows n = 0..100, flattened
• Dennis Walsh, Notes on antisymmetric subsets of {1,2,...,n}

Crossrefs

Cf. A108411, row sums of triangle T(n,k).

Cf. A000079, A007318, A013109, A152198

Programs

Maple

seq(seq(binomial(floor(n/2),k)*2^k,k=0..floor(n/2)),n=0..22);

Mathematica

Table[ CoefficientList[(1 + 2*x)^n, x] , {n, 0, 7}, {2}] // Flatten (* Jean-François Alcover, Aug 19 2013, after Philippe Deléham *)

Formula

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) = A152198(n,k)*2^k. - Philippe Deléham, Apr 09 2012

G.f.: (1+x)/(1-x^2-2*y*x^2). - Philippe Deléham, Apr 09 2012

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