Identifier
Identifier
Values
[1] => 1
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 1
[2,1] => 2
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 2
[1,3] => 1
[2,1,1] => 3
[2,2] => 3
[3,1] => 3
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 2
[1,1,3] => 1
[1,2,1,1] => 3
[1,2,2] => 3
[1,3,1] => 3
[1,4] => 1
[2,1,1,1] => 4
[2,1,2] => 4
[2,2,1] => 8
[2,3] => 4
[3,1,1] => 6
[3,2] => 6
[4,1] => 4
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 2
[1,1,1,3] => 1
[1,1,2,1,1] => 3
[1,1,2,2] => 3
[1,1,3,1] => 3
[1,1,4] => 1
[1,2,1,1,1] => 4
[1,2,1,2] => 4
[1,2,2,1] => 8
[1,2,3] => 4
[1,3,1,1] => 6
[1,3,2] => 6
[1,4,1] => 4
[1,5] => 1
[2,1,1,1,1] => 5
[2,1,1,2] => 5
[2,1,2,1] => 10
[2,1,3] => 5
[2,2,1,1] => 15
[2,2,2] => 15
[2,3,1] => 15
[2,4] => 5
[3,1,1,1] => 10
[3,1,2] => 10
[3,2,1] => 20
[3,3] => 10
[4,1,1] => 10
[4,2] => 10
[5,1] => 5
[6] => 1
Description
The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
References
[1] Berg, C., Bergeron, N., Saliola, F., Serrano, L., Zabrocki, M. The immaculate basis of the non-commutative symmetric functions arXiv:1303.4801
[2] Berg, C., Bergeron, N., Saliola, F., Serrano, L., Zabrocki, M. A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions arXiv:1208.5191
Code
def statistic(mu):
F = QuasiSymmetricFunctions(ZZ).F()
dI = QuasiSymmetricFunctions(ZZ).dI()
return sum(coeff for _, coeff in F(dI(mu)))

Created
Mar 24, 2013 at 23:08 by Chris Berg
Updated
May 20, 2017 at 22:31 by Martin Rubey