Refined restricted permutations. Derangements (and more generally the notion of ”fixed points of a permutation”) are concepts related to the cycle-structre, i.e. two-line notation, i.e. permutations qua 1-1 functions from [1,n] to [1,n]. On the other hand, Pattern-avoidance (and Wilf- equivalence) are inherently ”wordy”, i.e. pertain to permutations qua words. Perhaps this is why no one noticed the amazing and easily-stated fact that the number of 132-avoiding derangements equals the number of 321-avoiding derangements, and even more amazingly, that the same is true if you replace ”derangement” by ”with i fixed points”, for ANY $i$ between $0$ and $n$. This astounding fact was first discovered emprically by Aaron Robertson ...

References in zbMATH (referenced in 17 articles , 1 standard article )

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  1. Eu, Sen-Peng; Fu, Tung-Shan; Pan, Yeh-Jong; Ting, Chien-Tai: Two refined major-balance identities on 321-avoiding involutions (2015)
  2. Trongsiriwat, Wuttisak: Permutation statistics and multiple pattern avoidance (2015)
  3. Barnabei, Marilena; Bonetti, Flavio; Elizalde, Sergi; Silimbani, Matteo: Descent sets on 321-avoiding involutions and hook decompositions of partitions (2014)
  4. Bloom, Jonathan: A refinement of Wilf-equivalence for patterns of length 4 (2014)
  5. Miner, Sam; Pak, Igor: The shape of random pattern-avoiding permutations (2014)
  6. Elizalde, Sergi: Fixed points and excedances in restricted permutations (2012)
  7. Bousquet-Mélou, Mireille: Counting permutations with no long monotone subsequence via generating trees and the kernel method (2011)
  8. Bloom, Jonathan; Saracino, Dan: Another look at bijections for pattern-avoiding permutations (2010)
  9. Humphreys, Katherine: A history and a survey of lattice path enumeration (2010)
  10. Bloom, Jonathan; Saracino, Dan: On bijections for pattern-avoiding permutations (2009)
  11. Brignall, Robert; Huczynska, Sophie; Vatter, Vincent: Simple permutations and algebraic generating functions (2008)
  12. Deutsch, Emeric; Robertson, Aaron; Saracino, Dan: Refined restricted involutions (2007)
  13. Mansour, Toufik; Deng, Eva Y.P.; Du, Rosena R.X.: Dyck paths and restricted permutations (2006)
  14. Elizalde, Sergi; Pak, Igor: Bijections for refined restricted permutations (2004)
  15. Elizalde, Sergi; Deutsch, Emeric: A simple and unusual bijection for Dyck paths and its consequences (2003)
  16. Mansour, Toufik; Robertson, Aaron: Refined restricted permutations avoiding subsets of patterns of length three (2002)
  17. Robertson, Aaron; Saracino, Dan; Zeilberger, Doron: Refined restricted permutations (2002)