***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew <info@findstat.org> * * * * 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: St001816 ----------------------------------------------------------------------------- Collection: Standard tableaux ----------------------------------------------------------------------------- Description: Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]]. ----------------------------------------------------------------------------- References: [1] Garsia, A. M., Wallach, N. $r$-Qsym is free over Sym [[MathSciNet:2319171]] [2] Lafrenière, N. Eigenvalues of symmetrized shuffling operators [[MathSciNet:4098299]] [3] Reiner, V., Saliola, F., Welker, V. Spectra of symmetrized shuffling operators [[MathSciNet:3184410]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [] => 0 [[1]] => 1 [[1,2]] => 2 [[1],[2]] => 0 [[1,2,3]] => 3 [[1,3],[2]] => 0 [[1,2],[3]] => 1 [[1],[2],[3]] => 1 [[1,2,3,4]] => 4 [[1,3,4],[2]] => 0 [[1,2,4],[3]] => 1 [[1,2,3],[4]] => 2 [[1,3],[2,4]] => 0 [[1,2],[3,4]] => 1 [[1,4],[2],[3]] => 1 [[1,3],[2],[4]] => 0 [[1,2],[3],[4]] => 2 [[1],[2],[3],[4]] => 0 [[1,2,3,4,5]] => 5 [[1,3,4,5],[2]] => 0 [[1,2,4,5],[3]] => 1 [[1,2,3,5],[4]] => 2 [[1,2,3,4],[5]] => 3 [[1,3,5],[2,4]] => 0 [[1,2,5],[3,4]] => 1 [[1,3,4],[2,5]] => 0 [[1,2,4],[3,5]] => 1 [[1,2,3],[4,5]] => 2 [[1,4,5],[2],[3]] => 1 [[1,3,5],[2],[4]] => 0 [[1,2,5],[3],[4]] => 2 [[1,3,4],[2],[5]] => 0 [[1,2,4],[3],[5]] => 1 [[1,2,3],[4],[5]] => 3 [[1,4],[2,5],[3]] => 1 [[1,3],[2,5],[4]] => 0 [[1,2],[3,5],[4]] => 2 [[1,3],[2,4],[5]] => 0 [[1,2],[3,4],[5]] => 1 [[1,5],[2],[3],[4]] => 0 [[1,4],[2],[3],[5]] => 1 [[1,3],[2],[4],[5]] => 0 [[1,2],[3],[4],[5]] => 1 [[1],[2],[3],[4],[5]] => 1 [[1,2,3,4,5,6]] => 6 [[1,3,4,5,6],[2]] => 0 [[1,2,4,5,6],[3]] => 1 [[1,2,3,5,6],[4]] => 2 [[1,2,3,4,6],[5]] => 3 [[1,2,3,4,5],[6]] => 4 [[1,3,5,6],[2,4]] => 0 [[1,2,5,6],[3,4]] => 1 [[1,3,4,6],[2,5]] => 0 [[1,2,4,6],[3,5]] => 1 [[1,2,3,6],[4,5]] => 2 [[1,3,4,5],[2,6]] => 0 [[1,2,4,5],[3,6]] => 1 [[1,2,3,5],[4,6]] => 2 [[1,2,3,4],[5,6]] => 3 [[1,4,5,6],[2],[3]] => 1 [[1,3,5,6],[2],[4]] => 0 [[1,2,5,6],[3],[4]] => 2 [[1,3,4,6],[2],[5]] => 0 [[1,2,4,6],[3],[5]] => 1 [[1,2,3,6],[4],[5]] => 3 [[1,3,4,5],[2],[6]] => 0 [[1,2,4,5],[3],[6]] => 1 [[1,2,3,5],[4],[6]] => 2 [[1,2,3,4],[5],[6]] => 4 [[1,3,5],[2,4,6]] => 0 [[1,2,5],[3,4,6]] => 1 [[1,3,4],[2,5,6]] => 0 [[1,2,4],[3,5,6]] => 1 [[1,2,3],[4,5,6]] => 2 [[1,4,6],[2,5],[3]] => 1 [[1,3,6],[2,5],[4]] => 0 [[1,2,6],[3,5],[4]] => 2 [[1,3,6],[2,4],[5]] => 0 [[1,2,6],[3,4],[5]] => 1 [[1,4,5],[2,6],[3]] => 1 [[1,3,5],[2,6],[4]] => 0 [[1,2,5],[3,6],[4]] => 2 [[1,3,4],[2,6],[5]] => 0 [[1,2,4],[3,6],[5]] => 1 [[1,2,3],[4,6],[5]] => 3 [[1,3,5],[2,4],[6]] => 0 [[1,2,5],[3,4],[6]] => 1 [[1,3,4],[2,5],[6]] => 0 [[1,2,4],[3,5],[6]] => 1 [[1,2,3],[4,5],[6]] => 2 [[1,5,6],[2],[3],[4]] => 0 [[1,4,6],[2],[3],[5]] => 1 [[1,3,6],[2],[4],[5]] => 0 [[1,2,6],[3],[4],[5]] => 1 [[1,4,5],[2],[3],[6]] => 1 [[1,3,5],[2],[4],[6]] => 0 [[1,2,5],[3],[4],[6]] => 2 [[1,3,4],[2],[5],[6]] => 0 [[1,2,4],[3],[5],[6]] => 1 [[1,2,3],[4],[5],[6]] => 2 [[1,4],[2,5],[3,6]] => 1 [[1,3],[2,5],[4,6]] => 0 [[1,2],[3,5],[4,6]] => 2 [[1,3],[2,4],[5,6]] => 0 [[1,2],[3,4],[5,6]] => 1 [[1,5],[2,6],[3],[4]] => 0 [[1,4],[2,6],[3],[5]] => 1 [[1,3],[2,6],[4],[5]] => 0 [[1,2],[3,6],[4],[5]] => 1 [[1,4],[2,5],[3],[6]] => 1 [[1,3],[2,5],[4],[6]] => 0 [[1,2],[3,5],[4],[6]] => 2 [[1,3],[2,4],[5],[6]] => 0 [[1,2],[3,4],[5],[6]] => 1 [[1,6],[2],[3],[4],[5]] => 1 [[1,5],[2],[3],[4],[6]] => 0 [[1,4],[2],[3],[5],[6]] => 1 [[1,3],[2],[4],[5],[6]] => 0 [[1,2],[3],[4],[5],[6]] => 2 [[1],[2],[3],[4],[5],[6]] => 0 [[1,2,3,4,5,6,7]] => 7 [[1,3,4,5,6,7],[2]] => 0 [[1,2,4,5,6,7],[3]] => 1 [[1,2,3,5,6,7],[4]] => 2 [[1,2,3,4,6,7],[5]] => 3 [[1,2,3,4,5,7],[6]] => 4 [[1,2,3,4,5,6],[7]] => 5 [[1,3,5,6,7],[2,4]] => 0 [[1,2,5,6,7],[3,4]] => 1 [[1,3,4,6,7],[2,5]] => 0 [[1,2,4,6,7],[3,5]] => 1 [[1,2,3,6,7],[4,5]] => 2 [[1,3,4,5,7],[2,6]] => 0 [[1,2,4,5,7],[3,6]] => 1 [[1,2,3,5,7],[4,6]] => 2 [[1,2,3,4,7],[5,6]] => 3 [[1,3,4,5,6],[2,7]] => 0 [[1,2,4,5,6],[3,7]] => 1 [[1,2,3,5,6],[4,7]] => 2 [[1,2,3,4,6],[5,7]] => 3 [[1,2,3,4,5],[6,7]] => 4 [[1,4,5,6,7],[2],[3]] => 1 [[1,3,5,6,7],[2],[4]] => 0 [[1,2,5,6,7],[3],[4]] => 2 [[1,3,4,6,7],[2],[5]] => 0 [[1,2,4,6,7],[3],[5]] => 1 [[1,2,3,6,7],[4],[5]] => 3 [[1,3,4,5,7],[2],[6]] => 0 [[1,2,4,5,7],[3],[6]] => 1 [[1,2,3,5,7],[4],[6]] => 2 [[1,2,3,4,7],[5],[6]] => 4 [[1,3,4,5,6],[2],[7]] => 0 [[1,2,4,5,6],[3],[7]] => 1 [[1,2,3,5,6],[4],[7]] => 2 [[1,2,3,4,6],[5],[7]] => 3 [[1,2,3,4,5],[6],[7]] => 5 [[1,3,5,7],[2,4,6]] => 0 [[1,2,5,7],[3,4,6]] => 1 [[1,3,4,7],[2,5,6]] => 0 [[1,2,4,7],[3,5,6]] => 1 [[1,2,3,7],[4,5,6]] => 2 [[1,3,5,6],[2,4,7]] => 0 [[1,2,5,6],[3,4,7]] => 1 [[1,3,4,6],[2,5,7]] => 0 [[1,2,4,6],[3,5,7]] => 1 [[1,2,3,6],[4,5,7]] => 2 [[1,3,4,5],[2,6,7]] => 0 [[1,2,4,5],[3,6,7]] => 1 [[1,2,3,5],[4,6,7]] => 2 [[1,2,3,4],[5,6,7]] => 3 [[1,4,6,7],[2,5],[3]] => 1 [[1,3,6,7],[2,5],[4]] => 0 [[1,2,6,7],[3,5],[4]] => 2 [[1,3,6,7],[2,4],[5]] => 0 [[1,2,6,7],[3,4],[5]] => 1 [[1,4,5,7],[2,6],[3]] => 1 [[1,3,5,7],[2,6],[4]] => 0 [[1,2,5,7],[3,6],[4]] => 2 [[1,3,4,7],[2,6],[5]] => 0 [[1,2,4,7],[3,6],[5]] => 1 [[1,2,3,7],[4,6],[5]] => 3 [[1,3,5,7],[2,4],[6]] => 0 [[1,2,5,7],[3,4],[6]] => 1 [[1,3,4,7],[2,5],[6]] => 0 [[1,2,4,7],[3,5],[6]] => 1 [[1,2,3,7],[4,5],[6]] => 2 [[1,4,5,6],[2,7],[3]] => 1 [[1,3,5,6],[2,7],[4]] => 0 [[1,2,5,6],[3,7],[4]] => 2 [[1,3,4,6],[2,7],[5]] => 0 [[1,2,4,6],[3,7],[5]] => 1 [[1,2,3,6],[4,7],[5]] => 3 [[1,3,4,5],[2,7],[6]] => 0 [[1,2,4,5],[3,7],[6]] => 1 [[1,2,3,5],[4,7],[6]] => 2 [[1,2,3,4],[5,7],[6]] => 4 [[1,3,5,6],[2,4],[7]] => 0 [[1,2,5,6],[3,4],[7]] => 1 [[1,3,4,6],[2,5],[7]] => 0 [[1,2,4,6],[3,5],[7]] => 1 [[1,2,3,6],[4,5],[7]] => 2 [[1,3,4,5],[2,6],[7]] => 0 [[1,2,4,5],[3,6],[7]] => 1 [[1,2,3,5],[4,6],[7]] => 2 [[1,2,3,4],[5,6],[7]] => 3 [[1,5,6,7],[2],[3],[4]] => 0 [[1,4,6,7],[2],[3],[5]] => 1 [[1,3,6,7],[2],[4],[5]] => 0 [[1,2,6,7],[3],[4],[5]] => 1 [[1,4,5,7],[2],[3],[6]] => 1 [[1,3,5,7],[2],[4],[6]] => 0 [[1,2,5,7],[3],[4],[6]] => 2 [[1,3,4,7],[2],[5],[6]] => 0 [[1,2,4,7],[3],[5],[6]] => 1 [[1,2,3,7],[4],[5],[6]] => 2 [[1,4,5,6],[2],[3],[7]] => 1 [[1,3,5,6],[2],[4],[7]] => 0 [[1,2,5,6],[3],[4],[7]] => 2 [[1,3,4,6],[2],[5],[7]] => 0 [[1,2,4,6],[3],[5],[7]] => 1 [[1,2,3,6],[4],[5],[7]] => 3 [[1,3,4,5],[2],[6],[7]] => 0 [[1,2,4,5],[3],[6],[7]] => 1 [[1,2,3,5],[4],[6],[7]] => 2 [[1,2,3,4],[5],[6],[7]] => 3 [[1,4,6],[2,5,7],[3]] => 1 [[1,3,6],[2,5,7],[4]] => 0 [[1,2,6],[3,5,7],[4]] => 2 [[1,3,6],[2,4,7],[5]] => 0 [[1,2,6],[3,4,7],[5]] => 1 [[1,4,5],[2,6,7],[3]] => 1 [[1,3,5],[2,6,7],[4]] => 0 [[1,2,5],[3,6,7],[4]] => 2 [[1,3,4],[2,6,7],[5]] => 0 [[1,2,4],[3,6,7],[5]] => 1 [[1,2,3],[4,6,7],[5]] => 3 [[1,3,5],[2,4,7],[6]] => 0 [[1,2,5],[3,4,7],[6]] => 1 [[1,3,4],[2,5,7],[6]] => 0 [[1,2,4],[3,5,7],[6]] => 1 [[1,2,3],[4,5,7],[6]] => 2 [[1,3,5],[2,4,6],[7]] => 0 [[1,2,5],[3,4,6],[7]] => 1 [[1,3,4],[2,5,6],[7]] => 0 [[1,2,4],[3,5,6],[7]] => 1 [[1,2,3],[4,5,6],[7]] => 2 [[1,4,7],[2,5],[3,6]] => 1 [[1,3,7],[2,5],[4,6]] => 0 [[1,2,7],[3,5],[4,6]] => 2 [[1,3,7],[2,4],[5,6]] => 0 [[1,2,7],[3,4],[5,6]] => 1 [[1,4,6],[2,5],[3,7]] => 1 [[1,3,6],[2,5],[4,7]] => 0 [[1,2,6],[3,5],[4,7]] => 2 [[1,3,6],[2,4],[5,7]] => 0 [[1,2,6],[3,4],[5,7]] => 1 [[1,4,5],[2,6],[3,7]] => 1 [[1,3,5],[2,6],[4,7]] => 0 [[1,2,5],[3,6],[4,7]] => 2 [[1,3,4],[2,6],[5,7]] => 0 [[1,2,4],[3,6],[5,7]] => 1 [[1,2,3],[4,6],[5,7]] => 3 [[1,3,5],[2,4],[6,7]] => 0 [[1,2,5],[3,4],[6,7]] => 1 [[1,3,4],[2,5],[6,7]] => 0 [[1,2,4],[3,5],[6,7]] => 1 [[1,2,3],[4,5],[6,7]] => 2 [[1,5,7],[2,6],[3],[4]] => 0 [[1,4,7],[2,6],[3],[5]] => 1 [[1,3,7],[2,6],[4],[5]] => 0 [[1,2,7],[3,6],[4],[5]] => 1 [[1,4,7],[2,5],[3],[6]] => 1 [[1,3,7],[2,5],[4],[6]] => 0 [[1,2,7],[3,5],[4],[6]] => 2 [[1,3,7],[2,4],[5],[6]] => 0 [[1,2,7],[3,4],[5],[6]] => 1 [[1,5,6],[2,7],[3],[4]] => 0 [[1,4,6],[2,7],[3],[5]] => 1 [[1,3,6],[2,7],[4],[5]] => 0 [[1,2,6],[3,7],[4],[5]] => 1 [[1,4,5],[2,7],[3],[6]] => 1 [[1,3,5],[2,7],[4],[6]] => 0 [[1,2,5],[3,7],[4],[6]] => 2 [[1,3,4],[2,7],[5],[6]] => 0 [[1,2,4],[3,7],[5],[6]] => 1 [[1,2,3],[4,7],[5],[6]] => 2 [[1,4,6],[2,5],[3],[7]] => 1 [[1,3,6],[2,5],[4],[7]] => 0 [[1,2,6],[3,5],[4],[7]] => 2 [[1,3,6],[2,4],[5],[7]] => 0 [[1,2,6],[3,4],[5],[7]] => 1 [[1,4,5],[2,6],[3],[7]] => 1 [[1,3,5],[2,6],[4],[7]] => 0 [[1,2,5],[3,6],[4],[7]] => 2 [[1,3,4],[2,6],[5],[7]] => 0 [[1,2,4],[3,6],[5],[7]] => 1 [[1,2,3],[4,6],[5],[7]] => 3 [[1,3,5],[2,4],[6],[7]] => 0 [[1,2,5],[3,4],[6],[7]] => 1 [[1,3,4],[2,5],[6],[7]] => 0 [[1,2,4],[3,5],[6],[7]] => 1 [[1,2,3],[4,5],[6],[7]] => 2 [[1,6,7],[2],[3],[4],[5]] => 1 [[1,5,7],[2],[3],[4],[6]] => 0 [[1,4,7],[2],[3],[5],[6]] => 1 [[1,3,7],[2],[4],[5],[6]] => 0 [[1,2,7],[3],[4],[5],[6]] => 2 [[1,5,6],[2],[3],[4],[7]] => 0 [[1,4,6],[2],[3],[5],[7]] => 1 [[1,3,6],[2],[4],[5],[7]] => 0 [[1,2,6],[3],[4],[5],[7]] => 1 [[1,4,5],[2],[3],[6],[7]] => 1 [[1,3,5],[2],[4],[6],[7]] => 0 [[1,2,5],[3],[4],[6],[7]] => 2 [[1,3,4],[2],[5],[6],[7]] => 0 [[1,2,4],[3],[5],[6],[7]] => 1 [[1,2,3],[4],[5],[6],[7]] => 3 [[1,5],[2,6],[3,7],[4]] => 0 [[1,4],[2,6],[3,7],[5]] => 1 [[1,3],[2,6],[4,7],[5]] => 0 [[1,2],[3,6],[4,7],[5]] => 1 [[1,4],[2,5],[3,7],[6]] => 1 [[1,3],[2,5],[4,7],[6]] => 0 [[1,2],[3,5],[4,7],[6]] => 2 [[1,3],[2,4],[5,7],[6]] => 0 [[1,2],[3,4],[5,7],[6]] => 1 [[1,4],[2,5],[3,6],[7]] => 1 [[1,3],[2,5],[4,6],[7]] => 0 [[1,2],[3,5],[4,6],[7]] => 2 [[1,3],[2,4],[5,6],[7]] => 0 [[1,2],[3,4],[5,6],[7]] => 1 [[1,6],[2,7],[3],[4],[5]] => 1 [[1,5],[2,7],[3],[4],[6]] => 0 [[1,4],[2,7],[3],[5],[6]] => 1 [[1,3],[2,7],[4],[5],[6]] => 0 [[1,2],[3,7],[4],[5],[6]] => 2 [[1,5],[2,6],[3],[4],[7]] => 0 [[1,4],[2,6],[3],[5],[7]] => 1 [[1,3],[2,6],[4],[5],[7]] => 0 [[1,2],[3,6],[4],[5],[7]] => 1 [[1,4],[2,5],[3],[6],[7]] => 1 [[1,3],[2,5],[4],[6],[7]] => 0 [[1,2],[3,5],[4],[6],[7]] => 2 [[1,3],[2,4],[5],[6],[7]] => 0 [[1,2],[3,4],[5],[6],[7]] => 1 [[1,7],[2],[3],[4],[5],[6]] => 0 [[1,6],[2],[3],[4],[5],[7]] => 1 [[1,5],[2],[3],[4],[6],[7]] => 0 [[1,4],[2],[3],[5],[6],[7]] => 1 [[1,3],[2],[4],[5],[6],[7]] => 0 [[1,2],[3],[4],[5],[6],[7]] => 1 [[1],[2],[3],[4],[5],[6],[7]] => 1 ----------------------------------------------------------------------------- Created: Jul 13, 2022 at 17:16 by Nadia Lafreniere ----------------------------------------------------------------------------- Last Updated: Jul 13, 2022 at 17:16 by Nadia Lafreniere