***************************************************************************** * 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: St001997 ----------------------------------------------------------------------------- Collection: Perfect matchings ----------------------------------------------------------------------------- Description: The number of edges in common with the identity matching. Can be seen as an analogue of counting the number of fixed points of a permutation [[St000022]]. After fixing one perfect matching as the identity matching, we can count the number of edges that a perfect matching has in common with the identity matching. Usually, one chooses the matching $12, 34, 56, \ldots$ as the identity matching. A deranged matching is a matching that has no edges in common with the identity matching. Corresponds to counting derangements in the symmetric group. The problem of counting deranged matchings was posed in [1] and solved in [2]. For more information on deranged matchings, see [3]. ----------------------------------------------------------------------------- References: [1] Brawner, J. N. Dinner, Dancing, and Tennis, Anyone? [[MathSciNet:1573426]] [2] Margolius, B. H. Avoiding your spouse at a bridge party [[MathSciNet:2075960]] [3] Johnston, D., Kayll, P. M., Palmer, C. Deranged matchings: proofs and conjectures [[arXiv:2209.11319]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [(1,2)] => 1 [(1,2),(3,4)] => 2 [(1,3),(2,4)] => 0 [(1,4),(2,3)] => 0 [(1,2),(3,4),(5,6)] => 3 [(1,3),(2,4),(5,6)] => 1 [(1,4),(2,3),(5,6)] => 1 [(1,5),(2,3),(4,6)] => 0 [(1,6),(2,3),(4,5)] => 0 [(1,6),(2,4),(3,5)] => 0 [(1,5),(2,4),(3,6)] => 0 [(1,4),(2,5),(3,6)] => 0 [(1,3),(2,5),(4,6)] => 0 [(1,2),(3,5),(4,6)] => 1 [(1,2),(3,6),(4,5)] => 1 [(1,3),(2,6),(4,5)] => 0 [(1,4),(2,6),(3,5)] => 0 [(1,5),(2,6),(3,4)] => 1 [(1,6),(2,5),(3,4)] => 1 [(1,2),(3,4),(5,6),(7,8)] => 4 [(1,3),(2,4),(5,6),(7,8)] => 2 [(1,4),(2,3),(5,6),(7,8)] => 2 [(1,5),(2,3),(4,6),(7,8)] => 1 [(1,6),(2,3),(4,5),(7,8)] => 1 [(1,7),(2,3),(4,5),(6,8)] => 0 [(1,8),(2,3),(4,5),(6,7)] => 0 [(1,8),(2,4),(3,5),(6,7)] => 0 [(1,7),(2,4),(3,5),(6,8)] => 0 [(1,6),(2,4),(3,5),(7,8)] => 1 [(1,5),(2,4),(3,6),(7,8)] => 1 [(1,4),(2,5),(3,6),(7,8)] => 1 [(1,3),(2,5),(4,6),(7,8)] => 1 [(1,2),(3,5),(4,6),(7,8)] => 2 [(1,2),(3,6),(4,5),(7,8)] => 2 [(1,3),(2,6),(4,5),(7,8)] => 1 [(1,4),(2,6),(3,5),(7,8)] => 1 [(1,5),(2,6),(3,4),(7,8)] => 2 [(1,6),(2,5),(3,4),(7,8)] => 2 [(1,7),(2,5),(3,4),(6,8)] => 1 [(1,8),(2,5),(3,4),(6,7)] => 1 [(1,8),(2,6),(3,4),(5,7)] => 1 [(1,7),(2,6),(3,4),(5,8)] => 1 [(1,6),(2,7),(3,4),(5,8)] => 1 [(1,5),(2,7),(3,4),(6,8)] => 1 [(1,4),(2,7),(3,5),(6,8)] => 0 [(1,3),(2,7),(4,5),(6,8)] => 0 [(1,2),(3,7),(4,5),(6,8)] => 1 [(1,2),(3,8),(4,5),(6,7)] => 1 [(1,3),(2,8),(4,5),(6,7)] => 0 [(1,4),(2,8),(3,5),(6,7)] => 0 [(1,5),(2,8),(3,4),(6,7)] => 1 [(1,6),(2,8),(3,4),(5,7)] => 1 [(1,7),(2,8),(3,4),(5,6)] => 2 [(1,8),(2,7),(3,4),(5,6)] => 2 [(1,8),(2,7),(3,5),(4,6)] => 0 [(1,7),(2,8),(3,5),(4,6)] => 0 [(1,6),(2,8),(3,5),(4,7)] => 0 [(1,5),(2,8),(3,6),(4,7)] => 0 [(1,4),(2,8),(3,6),(5,7)] => 0 [(1,3),(2,8),(4,6),(5,7)] => 0 [(1,2),(3,8),(4,6),(5,7)] => 1 [(1,2),(3,7),(4,6),(5,8)] => 1 [(1,3),(2,7),(4,6),(5,8)] => 0 [(1,4),(2,7),(3,6),(5,8)] => 0 [(1,5),(2,7),(3,6),(4,8)] => 0 [(1,6),(2,7),(3,5),(4,8)] => 0 [(1,7),(2,6),(3,5),(4,8)] => 0 [(1,8),(2,6),(3,5),(4,7)] => 0 [(1,8),(2,5),(3,6),(4,7)] => 0 [(1,7),(2,5),(3,6),(4,8)] => 0 [(1,6),(2,5),(3,7),(4,8)] => 0 [(1,5),(2,6),(3,7),(4,8)] => 0 [(1,4),(2,6),(3,7),(5,8)] => 0 [(1,3),(2,6),(4,7),(5,8)] => 0 [(1,2),(3,6),(4,7),(5,8)] => 1 [(1,2),(3,5),(4,7),(6,8)] => 1 [(1,3),(2,5),(4,7),(6,8)] => 0 [(1,4),(2,5),(3,7),(6,8)] => 0 [(1,5),(2,4),(3,7),(6,8)] => 0 [(1,6),(2,4),(3,7),(5,8)] => 0 [(1,7),(2,4),(3,6),(5,8)] => 0 [(1,8),(2,4),(3,6),(5,7)] => 0 [(1,8),(2,3),(4,6),(5,7)] => 0 [(1,7),(2,3),(4,6),(5,8)] => 0 [(1,6),(2,3),(4,7),(5,8)] => 0 [(1,5),(2,3),(4,7),(6,8)] => 0 [(1,4),(2,3),(5,7),(6,8)] => 0 [(1,3),(2,4),(5,7),(6,8)] => 0 [(1,2),(3,4),(5,7),(6,8)] => 2 [(1,2),(3,4),(5,8),(6,7)] => 2 [(1,3),(2,4),(5,8),(6,7)] => 0 [(1,4),(2,3),(5,8),(6,7)] => 0 [(1,5),(2,3),(4,8),(6,7)] => 0 [(1,6),(2,3),(4,8),(5,7)] => 0 [(1,7),(2,3),(4,8),(5,6)] => 1 [(1,8),(2,3),(4,7),(5,6)] => 1 [(1,8),(2,4),(3,7),(5,6)] => 1 [(1,7),(2,4),(3,8),(5,6)] => 1 [(1,6),(2,4),(3,8),(5,7)] => 0 [(1,5),(2,4),(3,8),(6,7)] => 0 [(1,4),(2,5),(3,8),(6,7)] => 0 [(1,3),(2,5),(4,8),(6,7)] => 0 [(1,2),(3,5),(4,8),(6,7)] => 1 [(1,2),(3,6),(4,8),(5,7)] => 1 [(1,3),(2,6),(4,8),(5,7)] => 0 [(1,4),(2,6),(3,8),(5,7)] => 0 [(1,5),(2,6),(3,8),(4,7)] => 0 [(1,6),(2,5),(3,8),(4,7)] => 0 [(1,7),(2,5),(3,8),(4,6)] => 0 [(1,8),(2,5),(3,7),(4,6)] => 0 [(1,8),(2,6),(3,7),(4,5)] => 0 [(1,7),(2,6),(3,8),(4,5)] => 0 [(1,6),(2,7),(3,8),(4,5)] => 0 [(1,5),(2,7),(3,8),(4,6)] => 0 [(1,4),(2,7),(3,8),(5,6)] => 1 [(1,3),(2,7),(4,8),(5,6)] => 1 [(1,2),(3,7),(4,8),(5,6)] => 2 [(1,2),(3,8),(4,7),(5,6)] => 2 [(1,3),(2,8),(4,7),(5,6)] => 1 [(1,4),(2,8),(3,7),(5,6)] => 1 [(1,5),(2,8),(3,7),(4,6)] => 0 [(1,6),(2,8),(3,7),(4,5)] => 0 [(1,7),(2,8),(3,6),(4,5)] => 0 [(1,8),(2,7),(3,6),(4,5)] => 0 ----------------------------------------------------------------------------- Created: Sep 20, 2026 at 17:26 by Lukas Klawuhn ----------------------------------------------------------------------------- Last Updated: Sep 20, 2026 at 17:26 by Lukas Klawuhn