Number of pairs of partitions of n that are successors in reverse lexicographic order, but incomparable in dominance (natural, majorization) ordering.
(history;
published version)
Discussion
Sat Apr 07
17:46
Andrey Zabolotskiy: OK, thank you.
COMMENTS
Empirical: a(n) is the number of zeros in the subdiagonal of the 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
COMMENTS
Equivalently, a(n) is the number of zeros in the subdiagonal of the 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
Discussion
Fri Mar 30
23:34
John M. Campbell: @Zabolotskiy Thank you for your feedback. I should've noted that my proposed comments are "Empirical". I'll add cross-references in A248475 to the other two sequences. I think that the ordering of the basis elements according to the inverse lexicographic ordering is fairly standard.
Discussion
Thu Mar 29
13:44
Andrey Zabolotskiy: Nice. Three questions: 1. Are all three proposed comments proven? 2. Perhaps add cross-references to other two sequences? 3. Is the ordering of the basis elements corresponding to the inverse lexicographic ordering of the corresponding partitions somewhat canonical? If no, perhaps it would be better to say "the lower-triangular matrix" to eliminate the ambiguity.
COMMENTS
Equivalently, a(n) is the number of zeroes in the sub-diagonal of the 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