Identifier
Values
[(1,2)] => [2,1] => [2,1] => [1,1,0,0] => 1
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => 2
[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => 4
[(1,4),(2,3)] => [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => 4
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,1,4,2,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0] => 5
[(1,4),(2,3),(5,6)] => [4,3,2,1,6,5] => [3,2,4,1,6,5] => [1,1,1,0,0,1,0,0,1,1,0,0] => 5
[(1,5),(2,3),(4,6)] => [5,3,2,6,1,4] => [3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0] => 7
[(1,6),(2,3),(4,5)] => [6,3,2,5,4,1] => [3,2,5,4,6,1] => [1,1,1,0,0,1,1,0,0,1,0,0] => 7
[(1,6),(2,4),(3,5)] => [6,4,5,2,3,1] => [4,2,5,3,6,1] => [1,1,1,1,0,0,1,0,0,1,0,0] => 9
[(1,5),(2,4),(3,6)] => [5,4,6,2,1,3] => [4,2,5,1,6,3] => [1,1,1,1,0,0,1,0,0,1,0,0] => 9
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [1,1,1,1,0,0,1,0,0,1,0,0] => 9
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0] => 7
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => [1,1,0,0,1,1,1,0,0,1,0,0] => 5
[(1,2),(3,6),(4,5)] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0] => 5
[(1,3),(2,6),(4,5)] => [3,6,1,5,4,2] => [3,1,5,4,6,2] => [1,1,1,0,0,1,1,0,0,1,0,0] => 7
[(1,4),(2,6),(3,5)] => [4,6,5,1,3,2] => [4,1,5,3,6,2] => [1,1,1,1,0,0,1,0,0,1,0,0] => 9
[(1,5),(2,6),(3,4)] => [5,6,4,3,1,2] => [4,3,5,1,6,2] => [1,1,1,1,0,0,1,0,0,1,0,0] => 9
[(1,6),(2,5),(3,4)] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => [1,1,1,1,0,0,1,0,0,1,0,0] => 9
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => 4
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0] => 6
[(1,2),(3,6),(4,5),(7,8)] => [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0] => 6
[(1,2),(3,7),(4,5),(6,8)] => [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0] => 8
[(1,2),(3,8),(4,5),(6,7)] => [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0] => 8
[(1,2),(3,8),(4,6),(5,7)] => [2,1,8,6,7,4,5,3] => [2,1,6,4,7,5,8,3] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0] => 10
[(1,2),(3,7),(4,6),(5,8)] => [2,1,7,6,8,4,3,5] => [2,1,6,4,7,3,8,5] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0] => 10
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,6,3,7,4,8,5] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0] => 10
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,5,3,7,4,8,6] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0] => 8
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,7,5,8,6] => [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0] => 6
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,8,7,6,5] => [2,1,4,3,7,6,8,5] => [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0] => 6
[(1,2),(3,5),(4,8),(6,7)] => [2,1,5,8,3,7,6,4] => [2,1,5,3,7,6,8,4] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0] => 8
[(1,2),(3,6),(4,8),(5,7)] => [2,1,6,8,7,3,5,4] => [2,1,6,3,7,5,8,4] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0] => 10
[(1,2),(3,7),(4,8),(5,6)] => [2,1,7,8,6,5,3,4] => [2,1,6,5,7,3,8,4] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0] => 10
[(1,2),(3,8),(4,7),(5,6)] => [2,1,8,7,6,5,4,3] => [2,1,6,5,7,4,8,3] => [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0] => 10
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Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with area sequences $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with St000005The bounce statistic of a Dyck path. and St000006The dinv of a Dyck path.. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
to permutation
Description
Returns the fixed point free involution whose transpositions are the pairs in the perfect matching.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.