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

1,6

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

Table of n, a(n) for n=1..55.

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

Cf. A182988, A248476, A001475, A076276.

Sequence in context: A013950 A350842 A018550 * A035952 A335754 A105781

Adjacent sequences: A248472 A248473 A248474 * A248476 A248477 A248478

Keyword

nonn

Author

Wouter Meeussen, Oct 07 2014

Status

approved