OFFSET
1,2
COMMENTS
Empirical: a(n) is the sum of the greatest elements over all lexicographically greatest elements in all partitions in the canonical basis of the Temperley-Lieb algebra of order n. - John M. Campbell, Oct 17 2017
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1699
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
Index entries for linear recurrences with constant coefficients, signature (6,-9,3).
FORMULA
O.g.f.: (1-3*q+2*q^2)/(1-6*q+9*q^2-3*q^3) more generally, sum( n!/(n-d)!*q^d/prod((1-r*q),r=1..d), d=0..n)/sum( q^d/prod((1-r*q),r=1..d), d=0..n) where n=4
EXAMPLE
a(1) = 3 because x1-x2, x2-x3, x3-x4 are all of degree 1 and are killed by the differential operator d_x1+d_x2+d_x3+d_x4
For example, the canonical basis of the Temperley-Lieb algebra of order 3 is {{{-3, 1}, {-2, -1}, {2, 3}}, {{-3, 3}, {-2, 2}, {-1, 1}}, {{-3, 3}, {-2, -1}, {1, 2}}, {{-3, -2}, {-1, 1}, {2, 3}}, {{-3, -2}, {-1, 3}, {1, 2}}}, and the lexicographically greatest elements among all partitions in this basis are {2, 3}, {-1, 1}, {1, 2}, {2, 3}, and {1, 2}, with a(3) = 3+1+2+3+2 = 11. - John M. Campbell, Oct 17 2017
MAPLE
coeffs(convert(series((1-3*q+2*q^2)/(1-6*q+9*q^2-3*q^3), q, 30), `+`)-O(q^30), q);
MATHEMATICA
LinearRecurrence[{6, -9, 3}, {1, 3, 11}, 24] (* Jean-François Alcover, Sep 22 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Mike Zabrocki, Aug 30 2006
STATUS
approved