A122368 Dimension of 4-variable non-commutative harmonics (twisted derivative). The dimension of the space of non-commutative polynomials in 4 variables which are killed by all symmetric differential operators (where for a monomial w, d_{xi} ( xi w ) = w and d_{xi} ( xj w ) = 0 for i/=j).

1, 3, 11, 42, 162, 627, 2430, 9423, 36549, 141777, 549990, 2133594, 8276985, 32109534, 124565121, 483235875, 1874657763, 7272519066, 28212902154, 109448714619, 424593725526, 1647162628047, 6389978382405, 24789187818585

Offset: 1

Author

Mike Zabrocki, Aug 30 2006

Keywords

• nonn

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

Examples

			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
		

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).

Crossrefs

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, A122370, A122371, A122372.

Programs

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 *)

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