The symmetric functions catalog

#Affine permutations

The set of affine permutations \(\symAff_n\) is the set of bijections \(\pi: \setZ \to \setZ\) satisfying \(\pi(i+n)=\pi(i)+n\) for all \(i\in \setZ\) and \(\pi(1)+\pi(2)+\dotsb+\pi(n)=1+2+\dotsb+n.\)

#Cayley permutations

A Cayley permutation of length \(n\) is a word \(w=(w_1,\dotsc,w_n)\) of positive integers whose set of values is \([k]\) for some \(k.\) Equivalently, these are packed words. They are enumerated by the ordered Bell, or Fubini, numbers A000670.

G. Cerbai and A. Claesson count fixed-point-free Cayley permutations using two-sort species and differential equations for the corresponding functional digraphs [CC25]. They obtain an explicit formula, prove that the fixed-point-free proportion tends to \(1/e,\) and also count the cases where the functional digraph is a tree, forest, or connected.

#Permutations of type \(B\)

The permutations of type \(B\) are the permutations of \(B_n\coloneqq\{\pm1,\pm2,\dotsc,\pm n\}\) satisfying \(\pi(-i)=-\pi(i).\) They form the hyperoctahedral group, which has order \(2^n n!\); see A000165.

E. Bagno, R. Biagioli, F. Jouhet, and Y. Roichman study fully commutative elements in type \(B\) through block number and descents [BBJR22]. They prove a Schur-positivity result for the corresponding descent enumerators.

#Colored permutations

A \(r\)-colored permutation of size \(n\) is a permutation in which each letter is assigned one of \(r\) colors. Equivalently, it is a pair \[(\pi,c) \in \symS_n \times \{0,1,\dotsc,r-1\}^n,\] usually written in colored window notation as \[\pi(1)^{(c_1)}\,\pi(2)^{(c_2)}\dotsm \pi(n)^{(c_n)}.\] The set of all such objects is the wreath product \[C_r \wr \symS_n \cong (\setZ/r\setZ)^n \rtimes \symS_n.\] In particular, there are \(r^n n!\) colored permutations of size \(n.\) For example, \[3^{(2)}\,1^{(0)}\,2^{(1)}\] is a \(3\)-colored permutation of size \(3.\)

When \(r=1\) we recover ordinary permutations; when \(r=2\) we recover the signed permutations, or type \(B\) permutations, after identifying the two colors with signs. Many permutation statistics, including descents and major index, have colored analogues; see [GG79, Hya12]. Colored permutations also appear in wreath-product Schur theory and in ribbon-tableau versions of RSK.

#Stirling permutations

In [GS78], Ira Gessel and Richard P. Stanley introduced the set of Stirling permutations, \(Q_n\). These are permutations of the multiset \(\{1,1,2,2,3,3,\dotsc,n,n\}\) with the additional property that between the two entries equal to \(i \in [n],\) only entries larger than \(i\) may appear. For example, \(12234431\) is an element in \(Q_n.\)

The cardinality is \(|Q_n|=(2n-1)!!\); see A001147.

Bibliography

1. [BBJR22]Eli Bagno, Riccardo Biagioli, Frédéric Jouhet and Yuval Roichman. Block number, descents and Schur positivity of fully commutative elements in $B_n$. European Journal of Combinatorics, 101:103464, 2022.
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   @article{BagnoBiagioliJouhetRoichman2022,
     author = {Eli Bagno and Riccardo Biagioli and Fr{\'e}d{\'e}ric Jouhet and
       Yuval Roichman},
     title = {Block number, descents and {S}chur positivity of fully commutative
       elements in {$B_n$}},
     year = {2022},
     journal = {European Journal of Combinatorics},
     volume = {101},
     pages = {103464},
     doi = {10.1016/j.ejc.2021.103464},
     eprint = {2012.06412}
   }
   

2. [CC25]Giulio Cerbai and Anders Claesson. Counting fixed-point-free Cayley permutations. arXiv:2507.09304, 2025.
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   @article{CerbaiClaesson2025x,
     author = {Giulio Cerbai and Anders Claesson},
     title = {Counting fixed-point-free {C}ayley permutations},
     year = {2025},
     eprint = {2507.09304},
     url = {https://arxiv.org/abs/2507.09304},
     doi = {10.1017/S0013091526101436},
     journal = {arXiv e-prints}
   }
   

3. [GG79]A. M Garsia and I. Gessel. Permutation statistics and partitions. Advances in Mathematics, 31(3):288–305, March 1979.
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   @article{GarsiaGessel1979,
     doi = {10.1016/0001-8708(79)90046-x},
     url2 = {https://doi.org/10.1016/0001-8708(79)90046-x},
     year = {1979},
     month = mar,
     publisher = {Elsevier {BV}},
     volume = {31},
     number = {3},
     pages = {288--305},
     author = {A.M Garsia and I. Gessel},
     title = {Permutation statistics and partitions},
     journal = {Advances in Mathematics}
   }
   

4. [GS78]Ira Gessel and Richard P Stanley. Stirling polynomials. Journal of Combinatorial Theory, Series A, 24(1):24–33, January 1978.
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   @article{GesselStanley1978,
     doi = {10.1016/0097-3165(78)90042-0},
     url2 = {https://doi.org/10.1016/0097-3165(78)90042-0},
     year = {1978},
     month = jan,
     publisher = {Elsevier {BV}},
     volume = {24},
     number = {1},
     pages = {24--33},
     author = {Ira Gessel and Richard P Stanley},
     title = {Stirling polynomials},
     journal = {Journal of Combinatorial Theory,  Series A}
   }
   

5. [Hya12]Matthew Hyatt. Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups. Advances in Applied Mathematics, 48(3):465–505, March 2012.
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   @article{Hyatt2012,
     doi = {10.1016/j.aam.2011.11.005},
     url2 = {https://doi.org/10.1016/j.aam.2011.11.005},
     year  = {2012},
     month = mar,
     publisher = {Elsevier {BV}},
     volume = {48},
     number = {3},
     pages = {465--505},
     author = {Matthew Hyatt},
     title = {Eulerian quasisymmetric functions for the type {B} {C}oxeter group and other wreath product groups},
     journal = {Advances in Applied Mathematics}
   }