cp's OEIS Frontend

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

%I A248475 #23 Apr 10 2018 14:29:38
%S A248475 0,0,0,0,0,2,3,4,6,9,12,17,22,30,39,51,65,85,107,136,171,216,268,335,
%T A248475 413,512,629,772,941,1151,1396,1694,2046,2471,2969,3569,4271,5110,
%U A248475 6093,7258,8620,10235,12113,14325,16902,19925,23434,27540,32296,37842,44260,51715,60322,70306,81805
%N A248475 Number of pairs of partitions of n that are successors in reverse lexicographic order, but incomparable in dominance (natural, majorization) ordering.
%C A248475 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
%D A248475 Ian G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1979, pp. 6-8.
%H A248475 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dominance_order">Dominance Order</a>
%e A248475 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.
%t A248475 Needs["Combinatorica`"];
%t A248475 dominant[par1_?PartitionQ,par2_?PartitionQ]:= Block[{le=Max[Length[par1],Length[par2]],acc},
%t A248475 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]];
%t A248475 Table[Count[Apply[dominant, Partition[Partitions[n], 2,1], 1],0], {n,40}]
%Y A248475 Cf. A182988, A248476, A001475, A076276.
%K A248475 nonn
%O A248475 1,6
%A A248475 _Wouter Meeussen_, Oct 07 2014