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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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
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.
Sequence in context: A076110 A117584 A199847 * A188211 A175009 A297395
KEYWORD
nonn,nice,tabl
AUTHOR
STATUS
approved

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Last modified February 22 22:43 EST 2024. Contains 370265 sequences. (Running on oeis4.)