Identifier
Values
[1] => [1] => [1] => ([],1) => 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2) => 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2) => 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,3,2] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,3,1] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[3,1,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[3,2,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,4,3] => [2,3,4,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,3,2,4] => [2,3,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[1,3,4,2] => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,4,2,3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
[1,4,3,2] => [3,4,2,1] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 1
[2,4,1,3] => [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[2,4,3,1] => [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[3,4,1,2] => [1,4,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[4,1,2,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
[4,2,1,3] => [3,1,4,2] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[4,2,3,1] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[4,3,1,2] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,3,5,4] => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[1,2,4,5,3] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 1
[1,3,4,5,2] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 1
[2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,3,5,4,1] => [5,2,3,4,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[3,4,5,1,2] => [1,5,2,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[3,5,2,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 1
[4,5,1,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 1
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[5,2,3,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,2,3,4,6,5] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[1,2,3,5,6,4] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[1,2,4,5,6,3] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[1,3,4,5,6,2] => [2,6,1,3,4,5] => [6,5,4,3,1,2] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[2,3,4,1,5,6] => [4,1,2,3,5,6] => [4,3,2,1,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[2,3,4,5,1,6] => [5,1,2,3,4,6] => [5,4,3,2,1,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[2,3,4,5,6,1] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,3,4,6,5,1] => [6,2,3,4,5,1] => [2,3,4,5,6,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[3,4,5,6,1,2] => [1,6,2,3,4,5] => [1,6,5,4,3,2] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[3,4,6,2,5,1] => [6,1,3,4,5,2] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[4,5,6,1,2,3] => [1,2,6,3,4,5] => [1,2,6,5,4,3] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[4,6,2,3,5,1] => [6,1,2,4,5,3] => [4,5,6,3,2,1] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[5,6,1,2,3,4] => [1,2,3,6,4,5] => [1,2,3,6,5,4] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[6,2,3,4,5,1] => [6,1,2,3,5,4] => [5,6,4,3,2,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => [8,7,6,5,4,3,2,1] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of minimal elements in a poset.
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.
Map
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
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.