

A122368


Dimension of 4variable noncommutative harmonics (twisted derivative). The dimension of the space of noncommutative 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).


5



1, 3, 11, 42, 162, 627, 2430, 9423, 36549, 141777, 549990, 2133594, 8276985, 32109534, 124565121, 483235875, 1874657763, 7272519066, 28212902154, 109448714619, 424593725526, 1647162628047, 6389978382405, 24789187818585
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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 TemperleyLieb algebra of order n.  John M. Campbell, Oct 17 2017


LINKS

Table of n, a(n) for n=1..24.
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, 266296.
C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778782.
Hanna Mularczyk, Lattice Paths and PatternAvoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626637.
Index entries for linear recurrences with constant coefficients, signature (6,9,3).


FORMULA

O.g.f.: (13*q+2*q^2)/(16*q+9*q^23*q^3) more generally, sum( n!/(nd)!*q^d/prod((1r*q),r=1..d), d=0..n)/sum( q^d/prod((1r*q),r=1..d), d=0..n) where n=4


EXAMPLE

a(1) = 3 because x1x2, x2x3, x3x4 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 TemperleyLieb 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((13*q+2*q^2)/(16*q+9*q^23*q^3), q, 30), `+`)O(q^30), q);


MATHEMATICA

LinearRecurrence[{6, 9, 3}, {1, 3, 11}, 24] (* JeanFrançois Alcover, Sep 22 2017 *)


CROSSREFS

Cf. A055105, A055107, A087903, A074664, A008277, A112340, A122367, A122369, A122370, A122371, A122372.
Sequence in context: A279704 A301483 A059716 * A032443 A180907 A143464
Adjacent sequences: A122365 A122366 A122367 * A122369 A122370 A122371


KEYWORD

nonn


AUTHOR

Mike Zabrocki, Aug 30 2006


STATUS

approved



