Identifier
Values
[(1,2)] => [2,1] => [2,1] => ([(0,1)],2) => 1
[(1,2),(3,4)] => [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
[(1,3),(2,4)] => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8) => 1
[(1,4),(2,3)] => [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
[(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(1,12),(2,6),(2,7),(2,12),(3,5),(3,7),(3,12),(5,9),(5,10),(6,9),(6,11),(7,9),(7,10),(7,11),(8,4),(9,13),(10,8),(10,13),(11,8),(11,13),(12,10),(12,11),(13,4)],14) => 1
[(1,3),(2,4),(5,6)] => [3,4,1,2,6,5] => [3,1,4,2,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,10),(1,15),(1,16),(2,11),(2,15),(2,16),(3,13),(3,15),(3,16),(4,12),(4,15),(4,16),(5,8),(5,9),(5,14),(5,20),(6,5),(6,10),(6,11),(6,12),(6,13),(8,17),(8,19),(9,17),(9,19),(10,18),(10,20),(11,14),(11,18),(11,20),(12,8),(12,18),(12,20),(13,9),(13,18),(13,20),(14,17),(14,19),(15,14),(15,20),(16,14),(16,18),(17,7),(18,19),(19,7),(20,17),(20,19)],21) => 1
[(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => [4,1,3,2,6,5] => ([(0,1),(0,3),(0,4),(0,5),(1,11),(1,14),(2,6),(2,8),(2,15),(3,10),(3,12),(3,14),(4,9),(4,10),(4,14),(5,2),(5,9),(5,11),(5,12),(6,17),(6,18),(8,17),(9,13),(9,15),(9,16),(10,13),(10,15),(11,8),(11,16),(12,6),(12,13),(12,16),(13,18),(14,15),(14,16),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7)],19) => 1
[(1,5),(2,3),(4,6)] => [3,5,2,6,1,4] => [5,1,3,2,6,4] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,21),(1,22),(2,8),(2,10),(2,17),(3,12),(3,13),(3,15),(3,17),(4,11),(4,14),(4,15),(4,17),(5,8),(5,9),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(8,19),(8,23),(9,1),(9,19),(9,20),(9,23),(10,18),(10,19),(11,18),(11,20),(12,18),(12,20),(12,23),(13,16),(13,23),(14,16),(14,20),(14,23),(15,16),(15,19),(15,20),(16,22),(17,18),(17,19),(17,23),(18,21),(19,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21),(23,22)],24) => 1
[(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => [6,1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(1,10),(1,15),(2,8),(2,11),(2,15),(3,7),(3,8),(3,9),(4,7),(4,10),(4,11),(4,15),(5,17),(7,12),(7,13),(7,16),(8,13),(8,16),(9,12),(9,16),(10,12),(10,14),(11,5),(11,13),(11,14),(12,18),(13,17),(13,18),(14,17),(14,18),(15,5),(15,14),(15,16),(16,17),(16,18),(17,6),(18,6)],19) => 1
[(1,2),(3,5),(4,6)] => [2,1,5,6,3,4] => [2,1,5,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,10),(1,15),(1,16),(2,11),(2,15),(2,16),(3,13),(3,15),(3,16),(4,12),(4,15),(4,16),(5,8),(5,9),(5,14),(5,20),(6,5),(6,10),(6,11),(6,12),(6,13),(8,17),(8,19),(9,17),(9,19),(10,18),(10,20),(11,14),(11,18),(11,20),(12,8),(12,18),(12,20),(13,9),(13,18),(13,20),(14,17),(14,19),(15,14),(15,20),(16,14),(16,18),(17,7),(18,19),(19,7),(20,17),(20,19)],21) => 1
[(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => [2,1,6,3,5,4] => ([(0,1),(0,3),(0,4),(0,5),(1,11),(1,15),(2,6),(2,8),(2,18),(3,12),(3,13),(3,15),(4,10),(4,13),(4,15),(5,2),(5,10),(5,11),(5,12),(6,16),(6,17),(7,16),(7,17),(8,16),(10,14),(10,18),(11,8),(11,18),(12,6),(12,14),(12,18),(13,7),(13,14),(14,17),(15,7),(15,18),(16,9),(17,9),(18,16),(18,17)],19) => 1
[(1,3),(2,6),(4,5)] => [3,5,1,6,4,2] => [3,1,6,2,5,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,16),(1,18),(2,8),(2,17),(2,19),(2,22),(3,11),(3,12),(3,18),(4,13),(4,14),(4,16),(4,18),(5,10),(5,11),(5,13),(5,16),(6,2),(6,10),(6,12),(6,14),(6,18),(7,20),(8,20),(8,21),(10,15),(10,19),(10,22),(11,15),(11,22),(12,15),(12,17),(12,19),(13,7),(13,19),(13,22),(14,8),(14,17),(14,19),(14,22),(15,21),(16,7),(16,22),(17,20),(17,21),(18,17),(18,22),(19,20),(19,21),(20,9),(21,9),(22,20),(22,21)],23) => 1
[(1,5),(2,6),(3,4)] => [4,5,6,3,1,2] => [6,2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22) => 1
[(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => [5,2,6,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(1,18),(1,21),(2,11),(2,13),(2,15),(3,12),(3,14),(3,15),(4,8),(4,10),(4,12),(4,15),(5,9),(5,10),(5,13),(5,15),(6,1),(6,8),(6,9),(6,11),(6,14),(8,17),(8,18),(8,19),(8,20),(9,17),(9,18),(9,19),(9,20),(10,20),(10,21),(11,17),(11,19),(12,18),(12,20),(13,17),(13,21),(14,18),(14,19),(14,21),(15,19),(15,20),(15,21),(16,7),(17,16),(17,22),(18,16),(18,22),(19,16),(19,22),(20,16),(20,22),(21,22),(22,7)],23) => 1
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searching the database for statistics with the same generating function
Description
The number of maximal elements of a poset.
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
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
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.