Identifier
Values
[1,0] => [(1,2)] => [2,1] => [2,1] => 1
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0] => [(1,4),(2,3)] => [3,4,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 2
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 2
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,5,3,4,2,1] => 1
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8)] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4
[1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 3
[1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 3
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 3
[1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 2
[1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 2
[1,1,1,1,0,0,0,0] => [(1,8),(2,7),(3,6),(4,5)] => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10)] => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5
[1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,10),(8,9)] => [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 4
[1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10)] => [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 4
[1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,10),(6,9),(7,8)] => [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 3
[1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,6),(4,5),(7,8),(9,10)] => [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 4
[1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,10),(8,9)] => [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 3
[1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10)] => [2,1,6,7,8,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => 3
[1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,10),(4,9),(5,8),(6,7)] => [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8),(9,10)] => [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 4
[1,1,0,0,1,0,1,1,0,0] => [(1,4),(2,3),(5,6),(7,10),(8,9)] => [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 3
[1,1,0,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10)] => [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 3
[1,1,0,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,10),(6,9),(7,8)] => [3,4,2,1,8,9,10,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 2
[1,1,1,0,0,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10)] => [4,5,6,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => 3
[1,1,1,0,0,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,10),(8,9)] => [4,5,6,3,2,1,9,10,8,7] => [6,5,4,3,2,1,10,9,8,7] => 2
[1,1,1,1,0,0,0,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10)] => [5,6,7,8,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 2
[1,1,1,1,1,0,0,0,0,0] => [(1,10),(2,9),(3,8),(4,7),(5,6)] => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)] => [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,4,3,6,5,8,7,12,11,10,9] => 5
[1,0,1,0,1,0,1,1,0,0,1,0] => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)] => [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,4,3,6,5,10,9,8,7,12,11] => 5
[1,0,1,0,1,0,1,1,1,0,0,0] => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)] => [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,4,3,6,5,12,11,10,9,8,7] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)] => [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,4,3,8,7,6,5,10,9,12,11] => 5
[1,0,1,0,1,1,0,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)] => [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,4,3,8,7,6,5,12,11,10,9] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] => [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,4,3,10,9,8,7,6,5,12,11] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)] => [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,4,3,12,11,10,9,8,7,6,5] => 3
[1,0,1,1,0,0,1,0,1,0,1,0] => [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)] => [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,6,5,4,3,8,7,10,9,12,11] => 5
[1,0,1,1,0,0,1,0,1,1,0,0] => [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)] => [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,6,5,4,3,8,7,12,11,10,9] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)] => [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,6,5,4,3,10,9,8,7,12,11] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)] => [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,6,5,4,3,12,11,10,9,8,7] => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)] => [2,1,6,7,8,5,4,3,10,9,12,11] => [2,1,8,7,6,5,4,3,10,9,12,11] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)] => [2,1,6,7,8,5,4,3,11,12,10,9] => [2,1,8,7,6,5,4,3,12,11,10,9] => 3
[1,0,1,1,1,1,0,0,0,0,1,0] => [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)] => [2,1,7,8,9,10,6,5,4,3,12,11] => [2,1,10,9,8,7,6,5,4,3,12,11] => 3
[1,0,1,1,1,1,1,0,0,0,0,0] => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)] => [2,1,8,9,10,11,12,7,6,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 2
[1,1,0,0,1,0,1,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)] => [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 5
[1,1,0,0,1,0,1,0,1,1,0,0] => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)] => [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 4
[1,1,0,0,1,0,1,1,0,0,1,0] => [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)] => [3,4,2,1,6,5,9,10,8,7,12,11] => [4,3,2,1,6,5,10,9,8,7,12,11] => 4
[1,1,0,0,1,0,1,1,1,0,0,0] => [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)] => [3,4,2,1,6,5,10,11,12,9,8,7] => [4,3,2,1,6,5,12,11,10,9,8,7] => 3
[1,1,0,0,1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)] => [3,4,2,1,7,8,6,5,10,9,12,11] => [4,3,2,1,8,7,6,5,10,9,12,11] => 4
[1,1,0,0,1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)] => [3,4,2,1,7,8,6,5,11,12,10,9] => [4,3,2,1,8,7,6,5,12,11,10,9] => 3
[1,1,0,0,1,1,1,0,0,0,1,0] => [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)] => [3,4,2,1,8,9,10,7,6,5,12,11] => [4,3,2,1,10,9,8,7,6,5,12,11] => 3
[1,1,0,0,1,1,1,1,0,0,0,0] => [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)] => [3,4,2,1,9,10,11,12,8,7,6,5] => [4,3,2,1,12,11,10,9,8,7,6,5] => 2
[1,1,1,0,0,0,1,0,1,0,1,0] => [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)] => [4,5,6,3,2,1,8,7,10,9,12,11] => [6,5,4,3,2,1,8,7,10,9,12,11] => 4
[1,1,1,0,0,0,1,0,1,1,0,0] => [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)] => [4,5,6,3,2,1,8,7,11,12,10,9] => [6,5,4,3,2,1,8,7,12,11,10,9] => 3
[1,1,1,0,0,0,1,1,0,0,1,0] => [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)] => [4,5,6,3,2,1,9,10,8,7,12,11] => [6,5,4,3,2,1,10,9,8,7,12,11] => 3
[1,1,1,0,0,0,1,1,1,0,0,0] => [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)] => [4,5,6,3,2,1,10,11,12,9,8,7] => [6,5,4,3,2,1,12,11,10,9,8,7] => 2
[1,1,1,1,0,0,0,0,1,0,1,0] => [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)] => [5,6,7,8,4,3,2,1,10,9,12,11] => [8,7,6,5,4,3,2,1,10,9,12,11] => 3
[1,1,1,1,0,0,0,0,1,1,0,0] => [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)] => [5,6,7,8,4,3,2,1,11,12,10,9] => [8,7,6,5,4,3,2,1,12,11,10,9] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)] => [6,7,8,9,10,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,12,11] => 2
[1,1,1,1,1,1,0,0,0,0,0,0] => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)] => [7,8,9,10,11,12,6,5,4,3,2,1] => [12,11,10,9,8,7,6,5,4,3,2,1] => 1
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Description
The number of big ascents of a permutation after prepending zero.
Given a permutation $\pi$ of $\{1,\ldots,n\}$ we set $\pi(0) = 0$ and then count the number of indices $i \in \{0,\ldots,n-1\}$ such that $\pi(i+1) - \pi(i) > 1$.
It was shown in [1, Theorem 1.3] and in [2, Corollary 5.7] that this statistic is equidistributed with the number of descents (St000021The number of descents of a permutation.).
G. Han provided a bijection on permutations sending this statistic to the number of descents [3] using a simple variant of the first fundamental transformation Mp00086first fundamental transformation.
St000646The number of big ascents of a permutation. is the statistic without the border condition $\pi(0) = 0$.
Map
Demazure product with inverse
Description
This map sends a permutation $\pi$ to $\pi^{-1} \star \pi$ where $\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations $\pi$ for which $\pi = \pi^{-1}$.
Map
to tunnel matching
Description
Sends a Dyck path of semilength n to the noncrossing perfect matching given by matching an up-step with the corresponding down-step.
This is, for a Dyck path $D$ of semilength $n$, the perfect matching of $\{1,\dots,2n\}$ with $i < j$ being matched if $D_i$ is an up-step and $D_j$ is the down-step connected to $D_i$ by a tunnel.
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.