A248475 Number of pairs of partitions of n that are successors in reverse lexicographic order, but incomparable in dominance (natural, majorization) ordering.

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

Author

Wouter Meeussen, Oct 07 2014

Keywords

• nonn

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

Examples

			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.
		

References

• Ian G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979, pp. 6-8.

Links

• Wikipedia, Dominance Order

Crossrefs

Cf. A182988, A248476, A001475, A076276.

Programs

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}]