Identifier
Values
[1] => [1,0,1,0] => [3,1,2] => [3,1,2] => 1
[2] => [1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => 3
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => 2
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,5,4,1,3] => 3
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => [4,1,3,2] => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,5,4,2] => 2
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,6,5,4,1,3] => 3
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [5,3,1,4,2] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [2,5,1,4,3] => 3
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,1,4,3,2] => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,6,5,4,2] => 2
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [6,3,5,1,4,2] => 3
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,5,1,4,3] => 3
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,5,4,2] => 2
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,5,3,2] => 2
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,1,5,4,3,2] => 2
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,6,5,1,4,3] => 3
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [4,3,1,6,5,2] => 2
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [2,6,5,1,4,3] => 3
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,4,3,2] => 2
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [3,1,6,5,4,2] => 2
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [2,6,1,5,4,3] => 3
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,6,4,3,2] => 2
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,6,5,1,4,3] => 3
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [6,4,1,5,3,2] => 2
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,1,6,5,4,2] => 2
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,3,1,6,4,2] => 2
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,1,5,4,3] => 3
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [6,1,5,4,3,2] => 2
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,6,5,3,2] => 2
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [6,3,1,5,4,2] => 2
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,6,1,5,4,3] => 3
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,1,5,4,3,2] => 2
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,6,1,5,4,3] => 3
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,6,5,4,2] => 2
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,1,5,4,3,2] => 2
[2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => 2
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,6,1,5,4,3] => 3
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,6,5,4,2] => 2
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [4,1,6,5,3,2] => 2
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,6,4,3,2] => 2
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,5,4,3,2] => 2
[3,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [7,1,6,5,2,3,8,4] => [7,1,8,6,5,4,3,2] => 2
[4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [7,1,8,6,2,3,4,5] => [7,1,8,6,5,4,3,2] => 2
[3,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [7,1,5,2,6,3,8,4] => [7,1,8,6,5,4,3,2] => 2
[5,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [7,1,8,5,2,3,4,6] => [7,1,8,6,5,4,3,2] => 2
[4,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,1,0,0,0] => [7,1,4,6,2,3,8,5] => [7,1,8,6,5,4,3,2] => 2
[4,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0] => [7,1,8,2,6,3,4,5] => [7,1,8,6,5,4,3,2] => 2
[3,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [7,1,2,5,6,3,8,4] => [7,1,8,6,5,4,3,2] => 2
[5,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [7,1,4,8,2,3,5,6] => [7,1,8,6,5,4,3,2] => 2
[4,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [7,1,4,2,6,3,8,5] => [7,1,8,6,5,4,3,2] => 2
[5,5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,1,0,0] => [7,1,4,5,2,3,8,6] => [7,1,8,6,5,4,3,2] => 2
[4,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0] => [7,1,2,6,3,4,8,5] => [7,1,8,6,5,4,3,2] => 2
[5,5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [7,1,5,2,3,4,8,6] => [7,1,8,6,5,4,3,2] => 2
[5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [7,1,2,8,3,4,5,6] => [7,1,8,6,5,4,3,2] => 2
[4,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [7,1,2,3,6,4,8,5] => [7,1,8,6,5,4,3,2] => 2
[] => [] => [1] => [1] => 0
[5,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => [7,1,8,6,5,4,3,2] => 2
[5,5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [7,1,2,5,3,4,8,6] => [7,1,8,6,5,4,3,2] => 2
[5,5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [7,1,4,2,3,5,8,6] => [7,1,8,6,5,4,3,2] => 2
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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
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.