A188440 - OEIS

%I #25 Dec 13 2015 07:17:20

%S 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,

%T 1,10,40,80,80,32,1,10,40,80,80,32,1,12,60,160,240,192,64,1,12,60,160,

%U 240,192,64,1,14,84,280,560,672,448,128,1,14,84,280

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

%C 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.)

%C 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.

%C Triangle, read by rows, A013609 rows repeated. - _Philippe Deléham_, Apr 09 2012

%C 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

%H T. D. Noe, <a href="/A188440/b188440.txt">Rows n = 0..100, flattened</a>

%H Dennis Walsh, <a href="http://frank.mtsu.edu/~dwalsh/ANTISYMM.pdf">Notes on antisymmetric subsets of {1,2,...,n}</a>

%F T(n,k) = 2^k*C(floor(n/2),k) where C(*,*) denotes a binomial coefficient.

%F Sum(T(n,k),k=0..floor(n/2)) = 3^floor(n/2) = A108411(n).

%F G.f. for columns(k fixed):(2t^2)^k/((1-t)*(1-t^2)^k).

%F T(n,k) = A152198(n,k)*2^k. - _Philippe Deléham_, Apr 09 2012

%F G.f.: (1+x)/(1-x^2-2*y*x^2). - _Philippe Deléham_, Apr 09 2012

%F 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

%e Triangle T(n,k) initial values 0 <= k <= floor(n/2), n=0..13:

%e   1

%e   1

%e   1   2

%e   1   2

%e   1   4   4

%e   1   4   4

%e   1   6  12   8

%e   1   6  12   8

%e   1   8  24  32  16

%e   1   8  24  32  16

%e   1  10  40  80  80  32

%e   1  10  40  80  80  32

%e   1  12  60 160 240 192  64

%e   1  12  60 160 240 192  64

%e   ...

%e For n=7 and k=2, T(7,2)=12 since there are 12 antisymmetric size-2 subsets of {1,2,...,7}:

%e   {1,2}, {1,3}, {1,5}, {1,6}, {2,3}, {2,5},

%e   {2,7}, {3,6}, {3,7}, {5,6}, {5,7}, and {6,7}.

%e (1, 0, -1, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, ...) begins:

%e   1

%e   1   0

%e   1   2   0

%e   1   2   0   0

%e   1   4   4   0   0

%e   1   4   4   0   0   0

%e   1   6  12   8   0   0   0

%e   1   6  12   8   0   0   0   0

%e   1   8  24  32  16   0   0   0   0

%e   1   8  24  32  16   0   0   0   0   0

%e   1  10  40  80  80  32   0   0   0   0   0

%e   1  10  40  80  80  32   0   0   0   0   0   0

%e   1  12  60 160 240 192  64   0   0   0   0   0   0

%e   1  12  60 160 240 192  64   0   0   0   0   0   0   0

%e - _Philippe Deléham_, Apr 09 2012

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

%t Table[ CoefficientList[(1 + 2*x)^n, x] , {n, 0, 7}, {2}] // Flatten (* _Jean-François Alcover_, Aug 19 2013, after _Philippe Deléham_ *)

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

%Y Cf. A000079, A007318, A013109, A152198

%K nice,easy,nonn,tabf

%O 0,4

%A _Dennis P. Walsh_, Mar 31 2011