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-----------------------------------------------------------------------------
Statistic identifier: St000811
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions.
For example, $p_{22} = s_{1111} - s_{211} + 2s_{22} - s_{31} + s_4$, so the statistic on the partition $22$ is 2.
This is also the sum of the character values at the given conjugacy class over all irreducible characters of the symmetric group. [2]
For a permutation $\pi$ of given cycle type, this is also the number of permutations whose square equals $\pi$. [2]
-----------------------------------------------------------------------------
References: [1] Sum of all entries in character table of the symmetric group S_n. [[OEIS:A082733]]
[2] Petrov, F. Roots of permutations [[MathOverflow:41784]]
-----------------------------------------------------------------------------
Code:
def statistic(mu):
s = SymmetricFunctions(ZZ).s()
p = SymmetricFunctions(ZZ).p()
return sum(coeff for _, coeff in s(p(mu)))
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 0
[1,1] => 2
[3] => 1
[2,1] => 0
[1,1,1] => 4
[4] => 0
[3,1] => 1
[2,2] => 2
[2,1,1] => 0
[1,1,1,1] => 10
[5] => 1
[4,1] => 0
[3,2] => 0
[3,1,1] => 2
[2,2,1] => 2
[2,1,1,1] => 0
[1,1,1,1,1] => 26
[6] => 0
[5,1] => 1
[4,2] => 0
[4,1,1] => 0
[3,3] => 4
[3,2,1] => 0
[3,1,1,1] => 4
[2,2,2] => 0
[2,2,1,1] => 4
[2,1,1,1,1] => 0
[1,1,1,1,1,1] => 76
[7] => 1
[6,1] => 0
[5,2] => 0
[5,1,1] => 2
[4,3] => 0
[4,2,1] => 0
[4,1,1,1] => 0
[3,3,1] => 4
[3,2,2] => 2
[3,2,1,1] => 0
[3,1,1,1,1] => 10
[2,2,2,1] => 0
[2,2,1,1,1] => 8
[2,1,1,1,1,1] => 0
[1,1,1,1,1,1,1] => 232
[8] => 0
[7,1] => 1
[6,2] => 0
[6,1,1] => 0
[5,3] => 1
[5,2,1] => 0
[5,1,1,1] => 4
[4,4] => 4
[4,3,1] => 0
[4,2,2] => 0
[4,2,1,1] => 0
[4,1,1,1,1] => 0
[3,3,2] => 0
[3,3,1,1] => 8
[3,2,2,1] => 2
[3,2,1,1,1] => 0
[3,1,1,1,1,1] => 26
[2,2,2,2] => 12
[2,2,2,1,1] => 0
[2,2,1,1,1,1] => 20
[2,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1] => 764
[9] => 1
[8,1] => 0
[7,2] => 0
[7,1,1] => 2
[6,3] => 0
[6,2,1] => 0
[6,1,1,1] => 0
[5,4] => 0
[5,3,1] => 1
[5,2,2] => 2
[5,2,1,1] => 0
[5,1,1,1,1] => 10
[4,4,1] => 4
[4,3,2] => 0
[4,3,1,1] => 0
[4,2,2,1] => 0
[4,2,1,1,1] => 0
[4,1,1,1,1,1] => 0
[3,3,3] => 10
[3,3,2,1] => 0
[3,3,1,1,1] => 16
[3,2,2,2] => 0
[3,2,2,1,1] => 4
[3,2,1,1,1,1] => 0
[3,1,1,1,1,1,1] => 76
[2,2,2,2,1] => 12
[2,2,2,1,1,1] => 0
[2,2,1,1,1,1,1] => 52
[2,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1] => 2620
[10] => 0
[9,1] => 1
[8,2] => 0
[8,1,1] => 0
[7,3] => 1
[7,2,1] => 0
[7,1,1,1] => 4
[6,4] => 0
[6,3,1] => 0
[6,2,2] => 0
[6,2,1,1] => 0
[6,1,1,1,1] => 0
[5,5] => 6
[5,4,1] => 0
[5,3,2] => 0
[5,3,1,1] => 2
[5,2,2,1] => 2
[5,2,1,1,1] => 0
[5,1,1,1,1,1] => 26
[4,4,2] => 0
[4,4,1,1] => 8
[4,3,3] => 0
[4,3,2,1] => 0
[4,3,1,1,1] => 0
[4,2,2,2] => 0
[4,2,2,1,1] => 0
[4,2,1,1,1,1] => 0
[4,1,1,1,1,1,1] => 0
[3,3,3,1] => 10
[3,3,2,2] => 8
[3,3,2,1,1] => 0
[3,3,1,1,1,1] => 40
[3,2,2,2,1] => 0
[3,2,2,1,1,1] => 8
[3,2,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => 232
[2,2,2,2,2] => 0
[2,2,2,2,1,1] => 24
[2,2,2,1,1,1,1] => 0
[2,2,1,1,1,1,1,1] => 152
[2,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1] => 9496
[11] => 1
[10,1] => 0
[9,2] => 0
[9,1,1] => 2
[8,3] => 0
[8,2,1] => 0
[8,1,1,1] => 0
[7,4] => 0
[7,3,1] => 1
[7,2,2] => 2
[7,2,1,1] => 0
[7,1,1,1,1] => 10
[6,5] => 0
[6,4,1] => 0
[6,3,2] => 0
[6,3,1,1] => 0
[6,2,2,1] => 0
[6,2,1,1,1] => 0
[6,1,1,1,1,1] => 0
[5,5,1] => 6
[5,4,2] => 0
[5,4,1,1] => 0
[5,3,3] => 4
[5,3,2,1] => 0
[5,3,1,1,1] => 4
[5,2,2,2] => 0
[5,2,2,1,1] => 4
[5,2,1,1,1,1] => 0
[5,1,1,1,1,1,1] => 76
[4,4,3] => 4
[4,4,2,1] => 0
[4,4,1,1,1] => 16
[4,3,3,1] => 0
[4,3,2,2] => 0
[4,3,2,1,1] => 0
[4,3,1,1,1,1] => 0
[4,2,2,2,1] => 0
[4,2,2,1,1,1] => 0
[4,2,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1] => 0
[3,3,3,2] => 0
[3,3,3,1,1] => 20
[3,3,2,2,1] => 8
[3,3,2,1,1,1] => 0
[3,3,1,1,1,1,1] => 104
[3,2,2,2,2] => 12
[3,2,2,2,1,1] => 0
[3,2,2,1,1,1,1] => 20
[3,2,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1] => 764
[2,2,2,2,2,1] => 0
[2,2,2,2,1,1,1] => 48
[2,2,2,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1] => 464
[2,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1] => 35696
[12] => 0
[11,1] => 1
[10,2] => 0
[10,1,1] => 0
[9,3] => 1
[9,2,1] => 0
[9,1,1,1] => 4
[8,4] => 0
[8,3,1] => 0
[8,2,2] => 0
[8,2,1,1] => 0
[8,1,1,1,1] => 0
[7,5] => 1
[7,4,1] => 0
[7,3,2] => 0
[7,3,1,1] => 2
[7,2,2,1] => 2
[7,2,1,1,1] => 0
[7,1,1,1,1,1] => 26
[6,6] => 6
[6,5,1] => 0
[6,4,2] => 0
[6,4,1,1] => 0
[6,3,3] => 0
[6,3,2,1] => 0
[6,3,1,1,1] => 0
[6,2,2,2] => 0
[6,2,2,1,1] => 0
[6,2,1,1,1,1] => 0
[6,1,1,1,1,1,1] => 0
[5,5,2] => 0
[5,5,1,1] => 12
[5,4,3] => 0
[5,4,2,1] => 0
[5,4,1,1,1] => 0
[5,3,3,1] => 4
[5,3,2,2] => 2
[5,3,2,1,1] => 0
[5,3,1,1,1,1] => 10
[5,2,2,2,1] => 0
[5,2,2,1,1,1] => 8
[5,2,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1] => 232
[4,4,4] => 0
[4,4,3,1] => 4
[4,4,2,2] => 8
[4,4,2,1,1] => 0
[4,4,1,1,1,1] => 40
[4,3,3,2] => 0
[4,3,3,1,1] => 0
[4,3,2,2,1] => 0
[4,3,2,1,1,1] => 0
[4,3,1,1,1,1,1] => 0
[4,2,2,2,2] => 0
[4,2,2,2,1,1] => 0
[4,2,2,1,1,1,1] => 0
[4,2,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1] => 0
[3,3,3,3] => 46
[3,3,3,2,1] => 0
[3,3,3,1,1,1] => 40
[3,3,2,2,2] => 0
[3,3,2,2,1,1] => 16
[3,3,2,1,1,1,1] => 0
[3,3,1,1,1,1,1,1] => 304
[3,2,2,2,2,1] => 12
[3,2,2,2,1,1,1] => 0
[3,2,2,1,1,1,1,1] => 52
[3,2,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1] => 2620
[2,2,2,2,2,2] => 120
[2,2,2,2,2,1,1] => 0
[2,2,2,2,1,1,1,1] => 120
[2,2,2,1,1,1,1,1,1] => 0
[2,2,1,1,1,1,1,1,1,1] => 1528
[2,1,1,1,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => 140152
[5,4,3,1] => 0
[5,4,2,2] => 0
[5,4,2,1,1] => 0
[5,3,3,2] => 0
[5,3,3,1,1] => 8
[5,3,2,2,1] => 2
[4,4,3,2] => 0
[4,4,3,1,1] => 8
[4,4,2,2,1] => 8
[4,3,3,2,1] => 0
[5,4,3,2] => 0
[5,4,3,1,1] => 0
[5,4,2,2,1] => 0
[5,3,3,2,1] => 0
[4,4,3,2,1] => 0
[5,4,3,2,1] => 0
-----------------------------------------------------------------------------
Created: May 20, 2017 at 17:50 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Mar 15, 2019 at 20:01 by Martin Rubey