A047997 Triangle of numbers a(n,k) = number of balance positions when k equal weights are placed at a k-subset of the points {-n, -(n-1), ..., n-1, n} on a centrally pivoted rod.

1, 1, 2, 1, 3, 5, 1, 4, 8, 12, 1, 5, 13, 24, 32, 1, 6, 18, 43, 73, 94, 1, 7, 25, 69, 141, 227, 289, 1, 8, 32, 104, 252, 480, 734, 910, 1, 9, 41, 150, 414, 920, 1656, 2430, 2934, 1, 10, 50, 207, 649, 1636, 3370, 5744, 8150, 9686, 1, 11, 61, 277, 967

Offset

1,3

Comments

Also the number of k-subsets of {1..2n-1} with mean n. - Gus Wiseman, Apr 16 2023

References

R. E. Odeh and E. J. Cockayne, Balancing weights on the integer line, J. Combin. Theory, 7 (1969), 130-135.

Links

Table of n, a(n) for n=1..60.

Formula

Equivalent to number of partitions of n(2k-n+1)/2 into up to n parts each no more than 2k-n+1 so a(n, k)=A067059(n, n(2k-n+1)/2); row sums are A047653(n)-1 = A212352(n). - Henry Bottomley, Aug 11 2001

Example

From Gus Wiseman, Apr 18 2023: (Start)
Triangle begins:
    1
    1    2
    1    3    5
    1    4    8   12
    1    5   13   24   32
    1    6   18   43   73   94
    1    7   25   69  141  227  289
    1    8   32  104  252  480  734  910
    1    9   41  150  414  920 1656 2430 2934
Row n = 4 counts the following balanced subsets:
  {0}  {-1,1}  {-1,0,1}   {-3,0,1,2}
       {-2,2}  {-2,0,2}   {-4,0,1,3}
       {-3,3}  {-3,0,3}   {-2,-1,0,3}
       {-4,4}  {-3,1,2}   {-2,-1,1,2}
               {-4,0,4}   {-3,-1,0,4}
               {-4,1,3}   {-3,-1,1,3}
               {-2,-1,3}  {-3,-2,1,4}
               {-3,-1,4}  {-3,-2,2,3}
                          {-4,-1,1,4}
                          {-4,-1,2,3}
                          {-4,-2,2,4}
                          {-4,-3,3,4}
(End)

Mathematica

a[n_, k_] := Length[ IntegerPartitions[ n*(2k - n + 1)/2, n, Range[2k - n + 1]]]; Flatten[ Table[ a[n, k], {k, 1, 11}, {n, 1, k}]] (* Jean-François Alcover, Jan 02 2012 *)
Table[Length[Select[Subsets[Range[-n, n]], Length[#]==k&&Total[#]==0&]], {n, 8}, {k, n}] (* Gus Wiseman, Apr 16 2023 *)

Crossrefs

Last column is a(n,n) = A002838(n).

Row sums are A212352(n) = A047653(n)-1 = A000980(n)/2-1.

A007318 counts subsets by length, A327481 by mean, A013580 by median.

A327475 counts subsets with integer mean.

Cf. A000975, A024718, A070925, A079309, A326512, A326513, A361801, A362046.

Sequence in context: A076110 A117584 A199847 * A188211 A175009 A297395

Adjacent sequences: A047994 A047995 A047996 * A047998 A047999 A048000

Keyword

nonn,nice,tabl

Author

N. J. A. Sloane

Status

approved