Identifier
Values
=>
Cc0002;cc-rep
[]=>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
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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]
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
[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)))
Created
May 20, 2017 at 17:50 by Martin Rubey
Updated
Mar 15, 2019 at 20:01 by Martin Rubey
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