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A248475
Number of pairs of partitions of n that are successors in reverse lexicographic order, but incomparable in dominance (natural, majorization) ordering.
3
0, 0, 0, 0, 0, 2, 3, 4, 6, 9, 12, 17, 22, 30, 39, 51, 65, 85, 107, 136, 171, 216, 268, 335, 413, 512, 629, 772, 941, 1151, 1396, 1694, 2046, 2471, 2969, 3569, 4271, 5110, 6093, 7258, 8620, 10235, 12113, 14325, 16902, 19925, 23434, 27540, 32296, 37842, 44260, 51715, 60322, 70306, 81805
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.
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