Identifier
Mp00283:
Perfect matchings
—non-nesting-exceedence permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Images
[(1,2)] => [2,1] => [2,1] => [2,1] => 0
[(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 000
[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => [3,4,1,2] => 000
[(1,4),(2,3)] => [3,4,2,1] => [4,1,3,2] => [2,4,3,1] => 000
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [4,3,5,2,6,1] => 00000
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,1,4,2,6,5] => [4,5,2,3,6,1] => 00000
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,1,3,2,6,5] => [3,5,4,2,6,1] => 00000
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [5,1,3,2,6,4] => [3,5,4,6,1,2] => 00000
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,1,3,2,5,4] => [3,5,2,6,4,1] => 00000
[(1,6),(2,4),(3,5)] => [4,5,6,2,3,1] => [6,1,4,2,5,3] => [3,5,6,1,4,2] => 00000
[(1,5),(2,4),(3,6)] => [4,5,6,2,1,3] => [5,1,4,2,6,3] => [3,5,6,2,1,4] => 00000
[(1,4),(2,5),(3,6)] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,5,6,1,2,3] => 00000
[(1,3),(2,5),(4,6)] => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [4,5,2,6,1,3] => 00000
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => [4,3,5,6,1,2] => 00000
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,3,5,4] => [4,3,2,6,5,1] => 00000
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [3,1,6,2,5,4] => [4,2,5,6,3,1] => 00000
[(1,4),(2,6),(3,5)] => [4,5,6,1,3,2] => [4,1,6,2,5,3] => [4,2,6,3,5,1] => 00000
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,2,5,1,4,3] => [2,4,6,5,3,1] => 00000
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [5,2,6,1,4,3] => [2,5,6,4,1,3] => 00000
[(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [5,4,6,3,7,2,8,1] => 0000000
[(1,3),(2,4),(5,6),(7,8)] => [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => [5,6,3,4,7,2,8,1] => 0000000
[(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [4,1,3,2,6,5,8,7] => [4,6,5,3,7,2,8,1] => 0000000
[(1,5),(2,3),(4,6),(7,8)] => [3,5,2,6,1,4,8,7] => [5,1,3,2,6,4,8,7] => [4,6,5,7,2,3,8,1] => 0000000
[(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => [6,1,3,2,5,4,8,7] => [4,6,3,7,5,2,8,1] => 0000000
[(1,7),(2,3),(4,5),(6,8)] => [3,5,2,7,4,8,1,6] => [7,1,3,2,5,4,8,6] => [4,6,3,7,5,8,1,2] => 0000000
[(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => [8,1,3,2,5,4,7,6] => [4,6,3,7,2,8,5,1] => 0000000
[(1,8),(2,4),(3,5),(6,7)] => [4,5,7,2,3,8,6,1] => [8,1,4,2,5,3,7,6] => [4,6,7,2,3,8,5,1] => 0000000
[(1,7),(2,4),(3,5),(6,8)] => [4,5,7,2,3,8,1,6] => [7,1,4,2,5,3,8,6] => [4,6,7,2,5,8,1,3] => 0000000
[(1,6),(2,4),(3,5),(7,8)] => [4,5,6,2,3,1,8,7] => [6,1,4,2,5,3,8,7] => [4,6,7,2,5,3,8,1] => 0000000
[(1,5),(2,4),(3,6),(7,8)] => [4,5,6,2,1,3,8,7] => [5,1,4,2,6,3,8,7] => [4,6,7,3,2,5,8,1] => 0000000
[(1,4),(2,5),(3,6),(7,8)] => [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => [5,6,7,2,3,4,8,1] => 0000000
[(1,3),(2,5),(4,6),(7,8)] => [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => [5,6,3,7,2,4,8,1] => 0000000
[(1,2),(3,5),(4,6),(7,8)] => [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => [5,4,6,7,2,3,8,1] => 0000000
[(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [2,1,6,3,5,4,8,7] => [5,4,3,7,6,2,8,1] => 0000000
[(1,3),(2,6),(4,5),(7,8)] => [3,5,1,6,4,2,8,7] => [3,1,6,2,5,4,8,7] => [5,3,6,7,4,2,8,1] => 0000000
[(1,4),(2,6),(3,5),(7,8)] => [4,5,6,1,3,2,8,7] => [4,1,6,2,5,3,8,7] => [5,3,7,4,6,2,8,1] => 0000000
[(1,5),(2,6),(3,4),(7,8)] => [4,5,6,3,1,2,8,7] => [6,2,5,1,4,3,8,7] => [3,5,7,6,4,2,8,1] => 0000000
[(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [5,2,6,1,4,3,8,7] => [3,6,7,5,2,4,8,1] => 0000000
[(1,7),(2,5),(3,4),(6,8)] => [4,5,7,3,2,8,1,6] => [5,2,7,1,4,3,8,6] => [3,6,7,5,2,8,1,4] => 0000000
[(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => [5,2,8,1,4,3,7,6] => [3,6,7,2,5,8,4,1] => 0000000
[(1,8),(2,6),(3,4),(5,7)] => [4,6,7,3,8,2,5,1] => [6,2,8,1,4,3,7,5] => [3,6,7,2,8,4,5,1] => 0000000
[(1,7),(2,6),(3,4),(5,8)] => [4,6,7,3,8,2,1,5] => [6,2,7,1,4,3,8,5] => [3,6,7,5,8,1,2,4] => 0000000
[(1,6),(2,7),(3,4),(5,8)] => [4,6,7,3,8,1,2,5] => [7,2,6,1,4,3,8,5] => [3,5,7,6,8,2,1,4] => 0000000
[(1,5),(2,7),(3,4),(6,8)] => [4,5,7,3,1,8,2,6] => [7,2,5,1,4,3,8,6] => [3,5,7,6,4,8,1,2] => 0000000
[(1,4),(2,7),(3,5),(6,8)] => [4,5,7,1,3,8,2,6] => [4,1,7,2,5,3,8,6] => [5,3,7,4,6,8,1,2] => 0000000
[(1,3),(2,7),(4,5),(6,8)] => [3,5,1,7,4,8,2,6] => [3,1,7,2,5,4,8,6] => [5,3,6,7,4,8,1,2] => 0000000
[(1,2),(3,7),(4,5),(6,8)] => [2,1,5,7,4,8,3,6] => [2,1,7,3,5,4,8,6] => [5,4,3,7,6,8,1,2] => 0000000
[(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => [2,1,8,3,5,4,7,6] => [5,4,3,7,2,8,6,1] => 0000000
[(1,3),(2,8),(4,5),(6,7)] => [3,5,1,7,4,8,6,2] => [3,1,8,2,5,4,7,6] => [5,3,6,7,2,8,4,1] => 0000000
[(1,4),(2,8),(3,5),(6,7)] => [4,5,7,1,3,8,6,2] => [4,1,8,2,5,3,7,6] => [5,3,7,4,2,8,6,1] => 0000000
[(1,5),(2,8),(3,4),(6,7)] => [4,5,7,3,1,8,6,2] => [8,2,5,1,4,3,7,6] => [3,5,7,6,2,8,4,1] => 0000000
[(1,6),(2,8),(3,4),(5,7)] => [4,6,7,3,8,1,5,2] => [8,2,6,1,4,3,7,5] => [3,5,7,6,8,1,4,2] => 0000000
[(1,7),(2,8),(3,4),(5,6)] => [4,6,7,3,8,5,1,2] => [7,1,4,3,8,2,6,5] => [4,2,7,6,8,5,1,3] => 0000000
[(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => [8,1,4,3,7,2,6,5] => [4,2,7,5,8,6,3,1] => 0000000
[(1,8),(2,7),(3,5),(4,6)] => [5,6,7,8,3,4,2,1] => [8,1,5,3,7,2,6,4] => [4,2,7,8,3,6,5,1] => 0000000
[(1,7),(2,8),(3,5),(4,6)] => [5,6,7,8,3,4,1,2] => [7,1,5,3,8,2,6,4] => [4,2,7,8,5,6,1,3] => 0000000
[(1,6),(2,8),(3,5),(4,7)] => [5,6,7,8,3,1,4,2] => [8,2,6,1,5,3,7,4] => [3,5,7,8,4,1,6,2] => 0000000
[(1,5),(2,8),(3,6),(4,7)] => [5,6,7,8,1,3,4,2] => [5,1,8,2,6,3,7,4] => [5,3,7,8,2,1,6,4] => 0000000
[(1,4),(2,8),(3,6),(5,7)] => [4,6,7,1,8,3,5,2] => [4,1,8,2,6,3,7,5] => [5,3,7,4,8,1,6,2] => 0000000
[(1,3),(2,8),(4,6),(5,7)] => [3,6,1,7,8,4,5,2] => [3,1,8,2,6,4,7,5] => [5,3,6,7,8,1,4,2] => 0000000
[(1,2),(3,8),(4,6),(5,7)] => [2,1,6,7,8,4,5,3] => [2,1,8,3,6,4,7,5] => [5,4,3,7,8,1,6,2] => 0000000
[(1,2),(3,7),(4,6),(5,8)] => [2,1,6,7,8,4,3,5] => [2,1,7,3,6,4,8,5] => [5,4,3,7,8,2,1,6] => 0000000
[(1,3),(2,7),(4,6),(5,8)] => [3,6,1,7,8,4,2,5] => [3,1,7,2,6,4,8,5] => [5,3,6,7,8,2,1,4] => 0000000
[(1,4),(2,7),(3,6),(5,8)] => [4,6,7,1,8,3,2,5] => [4,1,7,2,6,3,8,5] => [5,3,7,4,8,2,1,6] => 0000000
[(1,5),(2,7),(3,6),(4,8)] => [5,6,7,8,1,3,2,4] => [5,1,7,2,6,3,8,4] => [5,3,7,8,2,4,1,6] => 0000000
[(1,6),(2,7),(3,5),(4,8)] => [5,6,7,8,3,1,2,4] => [7,2,6,1,5,3,8,4] => [3,5,7,8,4,2,1,6] => 0000000
[(1,7),(2,6),(3,5),(4,8)] => [5,6,7,8,3,2,1,4] => [6,2,7,1,5,3,8,4] => [3,6,7,8,4,1,2,5] => 0000000
[(1,8),(2,6),(3,5),(4,7)] => [5,6,7,8,3,2,4,1] => [6,2,8,1,5,3,7,4] => [3,6,7,8,1,4,5,2] => 0000000
[(1,8),(2,5),(3,6),(4,7)] => [5,6,7,8,2,3,4,1] => [8,1,5,2,6,3,7,4] => [4,6,7,8,1,2,5,3] => 0000000
[(1,7),(2,5),(3,6),(4,8)] => [5,6,7,8,2,3,1,4] => [7,1,5,2,6,3,8,4] => [4,6,7,8,1,3,2,5] => 0000000
[(1,6),(2,5),(3,7),(4,8)] => [5,6,7,8,2,1,3,4] => [6,1,5,2,7,3,8,4] => [4,6,7,8,2,1,3,5] => 0000000
[(1,5),(2,6),(3,7),(4,8)] => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [5,6,7,8,1,2,3,4] => 0000000
[(1,4),(2,6),(3,7),(5,8)] => [4,6,7,1,8,2,3,5] => [4,1,6,2,7,3,8,5] => [5,6,7,2,8,1,3,4] => 0000000
[(1,3),(2,6),(4,7),(5,8)] => [3,6,1,7,8,2,4,5] => [3,1,6,2,7,4,8,5] => [5,6,3,7,8,1,2,4] => 0000000
[(1,2),(3,6),(4,7),(5,8)] => [2,1,6,7,8,3,4,5] => [2,1,6,3,7,4,8,5] => [5,4,6,7,8,1,2,3] => 0000000
[(1,2),(3,5),(4,7),(6,8)] => [2,1,5,7,3,8,4,6] => [2,1,5,3,7,4,8,6] => [5,4,6,7,2,8,1,3] => 0000000
[(1,3),(2,5),(4,7),(6,8)] => [3,5,1,7,2,8,4,6] => [3,1,5,2,7,4,8,6] => [5,6,3,7,2,8,1,4] => 0000000
[(1,4),(2,5),(3,7),(6,8)] => [4,5,7,1,2,8,3,6] => [4,1,5,2,7,3,8,6] => [5,6,7,2,3,8,1,4] => 0000000
[(1,5),(2,4),(3,7),(6,8)] => [4,5,7,2,1,8,3,6] => [5,1,4,2,7,3,8,6] => [4,6,7,3,2,8,1,5] => 0000000
[(1,6),(2,4),(3,7),(5,8)] => [4,6,7,2,8,1,3,5] => [6,1,4,2,7,3,8,5] => [4,6,7,3,8,1,2,5] => 0000000
[(1,7),(2,4),(3,6),(5,8)] => [4,6,7,2,8,3,1,5] => [7,1,4,2,6,3,8,5] => [4,6,7,2,8,3,1,5] => 0000000
[(1,8),(2,4),(3,6),(5,7)] => [4,6,7,2,8,3,5,1] => [8,1,4,2,6,3,7,5] => [4,6,7,2,8,1,5,3] => 0000000
[(1,8),(2,3),(4,6),(5,7)] => [3,6,2,7,8,4,5,1] => [8,1,3,2,6,4,7,5] => [4,6,3,7,8,1,5,2] => 0000000
[(1,7),(2,3),(4,6),(5,8)] => [3,6,2,7,8,4,1,5] => [7,1,3,2,6,4,8,5] => [4,6,3,7,8,2,1,5] => 0000000
[(1,6),(2,3),(4,7),(5,8)] => [3,6,2,7,8,1,4,5] => [6,1,3,2,7,4,8,5] => [4,6,5,7,8,1,2,3] => 0000000
[(1,5),(2,3),(4,7),(6,8)] => [3,5,2,7,1,8,4,6] => [5,1,3,2,7,4,8,6] => [4,6,5,7,2,8,1,3] => 0000000
[(1,4),(2,3),(5,7),(6,8)] => [3,4,2,1,7,8,5,6] => [4,1,3,2,7,5,8,6] => [4,6,5,3,7,8,1,2] => 0000000
[(1,3),(2,4),(5,7),(6,8)] => [3,4,1,2,7,8,5,6] => [3,1,4,2,7,5,8,6] => [5,6,3,4,7,8,1,2] => 0000000
[(1,2),(3,4),(5,7),(6,8)] => [2,1,4,3,7,8,5,6] => [2,1,4,3,7,5,8,6] => [5,4,6,3,7,8,1,2] => 0000000
[(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,5,7,6] => [5,4,6,3,2,8,7,1] => 0000000
[(1,3),(2,4),(5,8),(6,7)] => [3,4,1,2,7,8,6,5] => [3,1,4,2,8,5,7,6] => [5,6,3,4,2,8,7,1] => 0000000
[(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [4,1,3,2,8,5,7,6] => [4,6,5,3,2,8,7,1] => 0000000
[(1,5),(2,3),(4,8),(6,7)] => [3,5,2,7,1,8,6,4] => [5,1,3,2,8,4,7,6] => [4,6,5,2,7,8,3,1] => 0000000
[(1,6),(2,3),(4,8),(5,7)] => [3,6,2,7,8,1,5,4] => [6,1,3,2,8,4,7,5] => [4,6,5,2,8,3,7,1] => 0000000
[(1,7),(2,3),(4,8),(5,6)] => [3,6,2,7,8,5,1,4] => [8,4,7,1,3,2,6,5] => [3,7,2,5,8,6,4,1] => 0000000
[(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => [7,4,8,1,3,2,6,5] => [3,7,2,6,8,5,1,4] => 0000000
[(1,8),(2,4),(3,7),(5,6)] => [4,6,7,2,8,5,3,1] => [7,3,8,1,4,2,6,5] => [3,7,5,2,8,6,1,4] => 0000000
[(1,7),(2,4),(3,8),(5,6)] => [4,6,7,2,8,5,1,3] => [8,3,7,1,4,2,6,5] => [3,7,4,2,8,6,5,1] => 0000000
[(1,6),(2,4),(3,8),(5,7)] => [4,6,7,2,8,1,5,3] => [6,1,4,2,8,3,7,5] => [4,6,2,7,8,3,5,1] => 0000000
[(1,5),(2,4),(3,8),(6,7)] => [4,5,7,2,1,8,6,3] => [5,1,4,2,8,3,7,6] => [4,6,2,7,5,8,3,1] => 0000000
[(1,4),(2,5),(3,8),(6,7)] => [4,5,7,1,2,8,6,3] => [4,1,5,2,8,3,7,6] => [5,6,2,4,7,8,3,1] => 0000000
>>> Load all 127 entries. <<<Map
non-nesting-exceedence permutation
Description
The fixed-point-free permutation with deficiencies given by the perfect matching, no alignments and no inversions between exceedences.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given perfect matching.
Put differently, the exceedences form the unique non-nesting perfect matching whose openers coincide with those of the given 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.
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.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
This map sends the major index to the number of inversions.
Map
connectivity set
Description
The connectivity set of a permutation as a binary word.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.
According to [2], also known as the global ascent set.
The connectivity set is
$$C(\pi)=\{i\in [n-1] | \forall 1 \leq j \leq i < k \leq n : \pi(j) < \pi(k)\}.$$
For $n > 1$ it can also be described as the set of occurrences of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
The permutation is connected, when the connectivity set is empty.
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