Identifier
Identifier
Values
[] generating graphics... => 1
[1] generating graphics... => 1
[2] generating graphics... => 0
[1,1] generating graphics... => 2
[3] generating graphics... => 1
[2,1] generating graphics... => 0
[1,1,1] generating graphics... => 4
[4] generating graphics... => 0
[3,1] generating graphics... => 1
[2,2] generating graphics... => 2
[2,1,1] generating graphics... => 0
[1,1,1,1] generating graphics... => 10
[5] generating graphics... => 1
[4,1] generating graphics... => 0
[3,2] generating graphics... => 0
[3,1,1] generating graphics... => 2
[2,2,1] generating graphics... => 2
[2,1,1,1] generating graphics... => 0
[1,1,1,1,1] generating graphics... => 26
[6] generating graphics... => 0
[5,1] generating graphics... => 1
[4,2] generating graphics... => 0
[4,1,1] generating graphics... => 0
[3,3] generating graphics... => 4
[3,2,1] generating graphics... => 0
[3,1,1,1] generating graphics... => 4
[2,2,2] generating graphics... => 0
[2,2,1,1] generating graphics... => 4
[2,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1] generating graphics... => 76
[7] generating graphics... => 1
[6,1] generating graphics... => 0
[5,2] generating graphics... => 0
[5,1,1] generating graphics... => 2
[4,3] generating graphics... => 0
[4,2,1] generating graphics... => 0
[4,1,1,1] generating graphics... => 0
[3,3,1] generating graphics... => 4
[3,2,2] generating graphics... => 2
[3,2,1,1] generating graphics... => 0
[3,1,1,1,1] generating graphics... => 10
[2,2,2,1] generating graphics... => 0
[2,2,1,1,1] generating graphics... => 8
[2,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1] generating graphics... => 232
[8] generating graphics... => 0
[7,1] generating graphics... => 1
[6,2] generating graphics... => 0
[6,1,1] generating graphics... => 0
[5,3] generating graphics... => 1
[5,2,1] generating graphics... => 0
[5,1,1,1] generating graphics... => 4
[4,4] generating graphics... => 4
[4,3,1] generating graphics... => 0
[4,2,2] generating graphics... => 0
[4,2,1,1] generating graphics... => 0
[4,1,1,1,1] generating graphics... => 0
[3,3,2] generating graphics... => 0
[3,3,1,1] generating graphics... => 8
[3,2,2,1] generating graphics... => 2
[3,2,1,1,1] generating graphics... => 0
[3,1,1,1,1,1] generating graphics... => 26
[2,2,2,2] generating graphics... => 12
[2,2,2,1,1] generating graphics... => 0
[2,2,1,1,1,1] generating graphics... => 20
[2,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1] generating graphics... => 764
[9] generating graphics... => 1
[8,1] generating graphics... => 0
[7,2] generating graphics... => 0
[7,1,1] generating graphics... => 2
[6,3] generating graphics... => 0
[6,2,1] generating graphics... => 0
[6,1,1,1] generating graphics... => 0
[5,4] generating graphics... => 0
[5,3,1] generating graphics... => 1
[5,2,2] generating graphics... => 2
[5,2,1,1] generating graphics... => 0
[5,1,1,1,1] generating graphics... => 10
[4,4,1] generating graphics... => 4
[4,3,2] generating graphics... => 0
[4,3,1,1] generating graphics... => 0
[4,2,2,1] generating graphics... => 0
[4,2,1,1,1] generating graphics... => 0
[4,1,1,1,1,1] generating graphics... => 0
[3,3,3] generating graphics... => 10
[3,3,2,1] generating graphics... => 0
[3,3,1,1,1] generating graphics... => 16
[3,2,2,2] generating graphics... => 0
[3,2,2,1,1] generating graphics... => 4
[3,2,1,1,1,1] generating graphics... => 0
[3,1,1,1,1,1,1] generating graphics... => 76
[2,2,2,2,1] generating graphics... => 12
[2,2,2,1,1,1] generating graphics... => 0
[2,2,1,1,1,1,1] generating graphics... => 52
[2,1,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1,1] generating graphics... => 2620
[10] generating graphics... => 0
[9,1] generating graphics... => 1
[8,2] generating graphics... => 0
[8,1,1] generating graphics... => 0
[7,3] generating graphics... => 1
[7,2,1] generating graphics... => 0
[7,1,1,1] generating graphics... => 4
[6,4] generating graphics... => 0
[6,3,1] generating graphics... => 0
[6,2,2] generating graphics... => 0
[6,2,1,1] generating graphics... => 0
[6,1,1,1,1] generating graphics... => 0
[5,5] generating graphics... => 6
[5,4,1] generating graphics... => 0
[5,3,2] generating graphics... => 0
[5,3,1,1] generating graphics... => 2
[5,2,2,1] generating graphics... => 2
[5,2,1,1,1] generating graphics... => 0
[5,1,1,1,1,1] generating graphics... => 26
[4,4,2] generating graphics... => 0
[4,4,1,1] generating graphics... => 8
[4,3,3] generating graphics... => 0
[4,3,2,1] generating graphics... => 0
[4,3,1,1,1] generating graphics... => 0
[4,2,2,2] generating graphics... => 0
[4,2,2,1,1] generating graphics... => 0
[4,2,1,1,1,1] generating graphics... => 0
[4,1,1,1,1,1,1] generating graphics... => 0
[3,3,3,1] generating graphics... => 10
[3,3,2,2] generating graphics... => 8
[3,3,2,1,1] generating graphics... => 0
[3,3,1,1,1,1] generating graphics... => 40
[3,2,2,2,1] generating graphics... => 0
[3,2,2,1,1,1] generating graphics... => 8
[3,2,1,1,1,1,1] generating graphics... => 0
[3,1,1,1,1,1,1,1] generating graphics... => 232
[2,2,2,2,2] generating graphics... => 0
[2,2,2,2,1,1] generating graphics... => 24
[2,2,2,1,1,1,1] generating graphics... => 0
[2,2,1,1,1,1,1,1] generating graphics... => 152
[2,1,1,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 9496
[11] generating graphics... => 1
[10,1] generating graphics... => 0
[9,2] generating graphics... => 0
[9,1,1] generating graphics... => 2
[8,3] generating graphics... => 0
[8,2,1] generating graphics... => 0
[8,1,1,1] generating graphics... => 0
[7,4] generating graphics... => 0
[7,3,1] generating graphics... => 1
[7,2,2] generating graphics... => 2
[7,2,1,1] generating graphics... => 0
[7,1,1,1,1] generating graphics... => 10
[6,5] generating graphics... => 0
[6,4,1] generating graphics... => 0
[6,3,2] generating graphics... => 0
[6,3,1,1] generating graphics... => 0
[6,2,2,1] generating graphics... => 0
[6,2,1,1,1] generating graphics... => 0
[6,1,1,1,1,1] generating graphics... => 0
[5,5,1] generating graphics... => 6
[5,4,2] generating graphics... => 0
[5,4,1,1] generating graphics... => 0
[5,3,3] generating graphics... => 4
[5,3,2,1] generating graphics... => 0
[5,3,1,1,1] generating graphics... => 4
[5,2,2,2] generating graphics... => 0
[5,2,2,1,1] generating graphics... => 4
[5,2,1,1,1,1] generating graphics... => 0
[5,1,1,1,1,1,1] generating graphics... => 76
[4,4,3] generating graphics... => 4
[4,4,2,1] generating graphics... => 0
[4,4,1,1,1] generating graphics... => 16
[4,3,3,1] generating graphics... => 0
[4,3,2,2] generating graphics... => 0
[4,3,2,1,1] generating graphics... => 0
[4,3,1,1,1,1] generating graphics... => 0
[4,2,2,2,1] generating graphics... => 0
[4,2,2,1,1,1] generating graphics... => 0
[4,2,1,1,1,1,1] generating graphics... => 0
[4,1,1,1,1,1,1,1] generating graphics... => 0
[3,3,3,2] generating graphics... => 0
[3,3,3,1,1] generating graphics... => 20
[3,3,2,2,1] generating graphics... => 8
[3,3,2,1,1,1] generating graphics... => 0
[3,3,1,1,1,1,1] generating graphics... => 104
[3,2,2,2,2] generating graphics... => 12
[3,2,2,2,1,1] generating graphics... => 0
[3,2,2,1,1,1,1] generating graphics... => 20
[3,2,1,1,1,1,1,1] generating graphics... => 0
[3,1,1,1,1,1,1,1,1] generating graphics... => 764
[2,2,2,2,2,1] generating graphics... => 0
[2,2,2,2,1,1,1] generating graphics... => 48
[2,2,2,1,1,1,1,1] generating graphics... => 0
[2,2,1,1,1,1,1,1,1] generating graphics... => 464
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 35696
[12] generating graphics... => 0
[11,1] generating graphics... => 1
[10,2] generating graphics... => 0
[10,1,1] generating graphics... => 0
[9,3] generating graphics... => 1
[9,2,1] generating graphics... => 0
[9,1,1,1] generating graphics... => 4
[8,4] generating graphics... => 0
[8,3,1] generating graphics... => 0
[8,2,2] generating graphics... => 0
[8,2,1,1] generating graphics... => 0
[8,1,1,1,1] generating graphics... => 0
[7,5] generating graphics... => 1
[7,4,1] generating graphics... => 0
[7,3,2] generating graphics... => 0
[7,3,1,1] generating graphics... => 2
[7,2,2,1] generating graphics... => 2
[7,2,1,1,1] generating graphics... => 0
[7,1,1,1,1,1] generating graphics... => 26
[6,6] generating graphics... => 6
[6,5,1] generating graphics... => 0
[6,4,2] generating graphics... => 0
[6,4,1,1] generating graphics... => 0
[6,3,3] generating graphics... => 0
[6,3,2,1] generating graphics... => 0
[6,3,1,1,1] generating graphics... => 0
[6,2,2,2] generating graphics... => 0
[6,2,2,1,1] generating graphics... => 0
[6,2,1,1,1,1] generating graphics... => 0
[6,1,1,1,1,1,1] generating graphics... => 0
[5,5,2] generating graphics... => 0
[5,5,1,1] generating graphics... => 12
[5,4,3] generating graphics... => 0
[5,4,2,1] generating graphics... => 0
[5,4,1,1,1] generating graphics... => 0
[5,3,3,1] generating graphics... => 4
[5,3,2,2] generating graphics... => 2
[5,3,2,1,1] generating graphics... => 0
[5,3,1,1,1,1] generating graphics... => 10
[5,2,2,2,1] generating graphics... => 0
[5,2,2,1,1,1] generating graphics... => 8
[5,2,1,1,1,1,1] generating graphics... => 0
[5,1,1,1,1,1,1,1] generating graphics... => 232
[4,4,4] generating graphics... => 0
[4,4,3,1] generating graphics... => 4
[4,4,2,2] generating graphics... => 8
[4,4,2,1,1] generating graphics... => 0
[4,4,1,1,1,1] generating graphics... => 40
[4,3,3,2] generating graphics... => 0
[4,3,3,1,1] generating graphics... => 0
[4,3,2,2,1] generating graphics... => 0
[4,3,2,1,1,1] generating graphics... => 0
[4,3,1,1,1,1,1] generating graphics... => 0
[4,2,2,2,2] generating graphics... => 0
[4,2,2,2,1,1] generating graphics... => 0
[4,2,2,1,1,1,1] generating graphics... => 0
[4,2,1,1,1,1,1,1] generating graphics... => 0
[4,1,1,1,1,1,1,1,1] generating graphics... => 0
[3,3,3,3] generating graphics... => 46
[3,3,3,2,1] generating graphics... => 0
[3,3,3,1,1,1] generating graphics... => 40
[3,3,2,2,2] generating graphics... => 0
[3,3,2,2,1,1] generating graphics... => 16
[3,3,2,1,1,1,1] generating graphics... => 0
[3,3,1,1,1,1,1,1] generating graphics... => 304
[3,2,2,2,2,1] generating graphics... => 12
[3,2,2,2,1,1,1] generating graphics... => 0
[3,2,2,1,1,1,1,1] generating graphics... => 52
[3,2,1,1,1,1,1,1,1] generating graphics... => 0
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 2620
[2,2,2,2,2,2] generating graphics... => 120
[2,2,2,2,2,1,1] generating graphics... => 0
[2,2,2,2,1,1,1,1] generating graphics... => 120
[2,2,2,1,1,1,1,1,1] generating graphics... => 0
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 1528
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 140152
[5,4,3,1] generating graphics... => 0
[5,4,2,2] generating graphics... => 0
[5,4,2,1,1] generating graphics... => 0
[5,3,3,2] generating graphics... => 0
[5,3,3,1,1] generating graphics... => 8
[5,3,2,2,1] generating graphics... => 2
[4,4,3,2] generating graphics... => 0
[4,4,3,1,1] generating graphics... => 8
[4,4,2,2,1] generating graphics... => 8
[4,3,3,2,1] generating graphics... => 0
[5,4,3,2] generating graphics... => 0
[5,4,3,1,1] generating graphics... => 0
[5,4,2,2,1] generating graphics... => 0
[5,3,3,2,1] generating graphics... => 0
[4,4,3,2,1] generating graphics... => 0
[5,4,3,2,1] generating graphics... => 0
click to show generating function       
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)))
Created
May 20, 2017 at 17:50 by Martin Rubey
Updated
Mar 15, 2019 at 20:01 by Martin Rubey