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• Matching, $\mathcal{M}_n$
• Perfect Matching of 4k
• Perfect Matching of 2k
• with 2k at the end
• concordant, discordant, dissident
• concordant, discordant, dissident_greater, , dissident_smaller


• Simsun Permutation
In words
• o_des, e_des

In cycles
• o_exc, e_exc
• exc, aexc, fix


• Permutaion, $\mathcal{S}_n$
• the Eulerian numbers with k des
• burhat order

Multi-Set Permutation $\mathcal{S}_M$
• exc
• ap, lap



• Permutaion of type B, $\mathcal{S}^B_n$
In words
• $\pi(0)=0$, basc, des$_B$, suc
• $\pi(0)=0$, des$_A$, des$_B$, negative

In cycles
• exc, aexc, fix
• the type B Eulerian numbers with k des


• Stirling Permutation, $\mathcal{Q}_n$
• asc, des, plat
• ap, lap

(1)-Stirling Permutation, $\mathcal{Q}^{(1)}_n$
• asc, des, plat
• lap, ap, even


• Quasi-Stirling Permutation, $\bar{\mathcal{Q}}_n$
• asc, des, plat
• ap, lap


• Derangement, $\mathcal{D}_n$
In multi-set
• exc