OFFSET
1,6
COMMENTS
Empirical: a(n) is the number of zeros in the subdiagonal of the lower-triangular matrix of coefficients giving the expansion of degree-n complete homogeneous symmetric functions in the Schur basis of the algebra of symmetric functions. - John M. Campbell, Mar 18 2018
REFERENCES
Ian G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979, pp. 6-8.
LINKS
Wikipedia, Dominance Order
EXAMPLE
The successor pair (3,1,1,1) and (2,2,2) are incomparable in dominance ordering, and so are their transposes (4,1,1) and (3,3) and these are the two only pairs for n=6, hence a(6)=2.
MATHEMATICA
Needs["Combinatorica`"];
dominant[par1_?PartitionQ, par2_?PartitionQ]:= Block[{le=Max[Length[par1], Length[par2]], acc},
acc=Accumulate[PadRight[par1, le]]-Accumulate[PadRight[par2, le]]; Which[Min[acc]===0&&Max[acc]>=0, 1, Min[acc]<=0&&Max[acc]===0, -1, True, 0]];
Table[Count[Apply[dominant, Partition[Partitions[n], 2, 1], 1], 0], {n, 40}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Wouter Meeussen, Oct 07 2014
STATUS
approved