***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- 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