cp's OEIS Frontend

A318177 a(n) is the number of integer partitions of n for which the Kimberling index is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.

%I A318177 #26 Dec 18 2023 10:09:43
%S A318177 0,0,0,1,0,1,1,2,3,2,2,5,5,8,8,11,18,20,26,26,35,49,56,73,88,101,130,
%T A318177 148,182,207,260,310,385,455,579,657,800,910,1135,1310,1546,1763,2169,
%U A318177 2488,2936,3352,3962,4612,5435,6187,7370,8430,9951,11276,13236,15133,17624,20009,23551,26464
%N A318177 a(n) is the number of integer partitions of n for which the Kimberling index is equal to the index of the seaweed algebra formed by the integer partition paired with its weight.
%C A318177 The index of a Lie algebra, g, is an invariant of the Lie algebra defined by min(dim(Ker(B_f)) where the min is taken over all linear functionals f on g and B_f denotes the bilinear form f([_,_]) were [,] denotes the bracket multiplication on g.
%C A318177 For seaweed subalgebras of sl(n), which are Lie subalgebras of sl(n) whose matrix representations are parametrized by an ordered pair of compositions of n, the index can be determined from a corresponding graph called a meander.
%C A318177 a(n)>0 for n=4 and n>5. To see this: for n>0 congruent to 0 (mod 4), say 4k+4, take the partition of the form (2k+3,2k+1); for n congruent to 2 (mod 4) if n=6 take (4,4,1), if n=10 take (5,3,2), if n>10, say n=4k+10, take the partition (2k+7,2k-1,1,1,1,1); for n>1 congruent to 1 (mod 6), say n=6k+1, take the partition (2k+3,2k-1,2k-1); for n>5 congruent to 5 (mod 6), say n=6k+5, take the partition (2k+3,2k+3,2k-1); for n>3 congruent to 3 (mod 6), say n=6k-3, take the partition (2k+1,2,...,2) with 2k-2 2's.
%H A318177 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/315.pdf">4-Shadows in q-Series and the Kimberling Index</a>, Preprint, May 15, 2016.
%H A318177 V. Coll, M. Hyatt, C. Magnant, and H. Wang, <a href="http://dx.doi.org/10.4172/1736-4337.1000227">Meander graphs and Frobenius seaweed Lie algebras II</a>, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
%H A318177 V. Dergachev and A. Kirillov, <a href="https://www.emis.de/journals/JLT/vol.10_no.2/6.html">Index of Lie algebras of seaweed type</a>, J. Lie Theory 10 (2) (2000) 331-343.
%Y A318177 Cf. A318176, A318178, A237832, A318196, A318203
%K A318177 nonn
%O A318177 1,8
%A A318177 _Nick Mayers_ and _Melissa Mayers_, Aug 20 2018