Identifier
Values
[1,0] => [(1,2)] => [2,1] => 1 => 1
[1,0,1,0] => [(1,2),(3,4)] => [2,1,4,3] => 101 => 2
[1,1,0,0] => [(1,4),(2,3)] => [3,4,2,1] => 110 => 1
[1,0,1,0,1,0] => [(1,2),(3,4),(5,6)] => [2,1,4,3,6,5] => 10101 => 5
[1,0,1,1,0,0] => [(1,2),(3,6),(4,5)] => [2,1,5,6,4,3] => 10110 => 3
[1,1,0,0,1,0] => [(1,4),(2,3),(5,6)] => [3,4,2,1,6,5] => 11001 => 3
[1,1,0,1,0,0] => [(1,6),(2,3),(4,5)] => [3,5,2,6,4,1] => 11010 => 2
[1,1,1,0,0,0] => [(1,6),(2,5),(3,4)] => [4,5,6,3,2,1] => 11100 => 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] => 1010101 => 14
[1,0,1,0,1,1,0,0] => [(1,2),(3,4),(5,8),(6,7)] => [2,1,4,3,7,8,6,5] => 1010110 => 9
[1,0,1,1,0,0,1,0] => [(1,2),(3,6),(4,5),(7,8)] => [2,1,5,6,4,3,8,7] => 1011001 => 10
[1,0,1,1,0,1,0,0] => [(1,2),(3,8),(4,5),(6,7)] => [2,1,5,7,4,8,6,3] => 1011010 => 7
[1,0,1,1,1,0,0,0] => [(1,2),(3,8),(4,7),(5,6)] => [2,1,6,7,8,5,4,3] => 1011100 => 4
[1,1,0,0,1,0,1,0] => [(1,4),(2,3),(5,6),(7,8)] => [3,4,2,1,6,5,8,7] => 1100101 => 9
[1,1,0,0,1,1,0,0] => [(1,4),(2,3),(5,8),(6,7)] => [3,4,2,1,7,8,6,5] => 1100110 => 6
[1,1,0,1,0,0,1,0] => [(1,6),(2,3),(4,5),(7,8)] => [3,5,2,6,4,1,8,7] => 1101001 => 7
[1,1,0,1,0,1,0,0] => [(1,8),(2,3),(4,5),(6,7)] => [3,5,2,7,4,8,6,1] => 1101010 => 5
[1,1,0,1,1,0,0,0] => [(1,8),(2,3),(4,7),(5,6)] => [3,6,2,7,8,5,4,1] => 1101100 => 3
[1,1,1,0,0,0,1,0] => [(1,6),(2,5),(3,4),(7,8)] => [4,5,6,3,2,1,8,7] => 1110001 => 4
[1,1,1,0,0,1,0,0] => [(1,8),(2,5),(3,4),(6,7)] => [4,5,7,3,2,8,6,1] => 1110010 => 3
[1,1,1,0,1,0,0,0] => [(1,8),(2,7),(3,4),(5,6)] => [4,6,7,3,8,5,2,1] => 1110100 => 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] => 1111000 => 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] => 101010101 => 42
[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] => 101010110 => 28
[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] => 101011001 => 32
[1,0,1,0,1,1,0,1,0,0] => [(1,2),(3,4),(5,10),(6,7),(8,9)] => [2,1,4,3,7,9,6,10,8,5] => 101011010 => 23
[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] => 101011100 => 14
[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] => 101100101 => 32
[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] => 101100110 => 22
[1,0,1,1,0,1,0,0,1,0] => [(1,2),(3,8),(4,5),(6,7),(9,10)] => [2,1,5,7,4,8,6,3,10,9] => 101101001 => 26
[1,0,1,1,0,1,0,1,0,0] => [(1,2),(3,10),(4,5),(6,7),(8,9)] => [2,1,5,7,4,9,6,10,8,3] => 101101010 => 19
[1,0,1,1,0,1,1,0,0,0] => [(1,2),(3,10),(4,5),(6,9),(7,8)] => [2,1,5,8,4,9,10,7,6,3] => 101101100 => 12
[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] => 101110001 => 17
[1,0,1,1,1,0,0,1,0,0] => [(1,2),(3,10),(4,7),(5,6),(8,9)] => [2,1,6,7,9,5,4,10,8,3] => 101110010 => 13
[1,0,1,1,1,0,1,0,0,0] => [(1,2),(3,10),(4,9),(5,6),(7,8)] => [2,1,6,8,9,5,10,7,4,3] => 101110100 => 9
[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] => 101111000 => 5
[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] => 110010101 => 28
[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] => 110010110 => 19
[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] => 110011001 => 22
[1,1,0,0,1,1,0,1,0,0] => [(1,4),(2,3),(5,10),(6,7),(8,9)] => [3,4,2,1,7,9,6,10,8,5] => 110011010 => 16
[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] => 110011100 => 10
[1,1,0,1,0,0,1,0,1,0] => [(1,6),(2,3),(4,5),(7,8),(9,10)] => [3,5,2,6,4,1,8,7,10,9] => 110100101 => 23
[1,1,0,1,0,0,1,1,0,0] => [(1,6),(2,3),(4,5),(7,10),(8,9)] => [3,5,2,6,4,1,9,10,8,7] => 110100110 => 16
[1,1,0,1,0,1,0,0,1,0] => [(1,8),(2,3),(4,5),(6,7),(9,10)] => [3,5,2,7,4,8,6,1,10,9] => 110101001 => 19
[1,1,0,1,0,1,0,1,0,0] => [(1,10),(2,3),(4,5),(6,7),(8,9)] => [3,5,2,7,4,9,6,10,8,1] => 110101010 => 14
[1,1,0,1,0,1,1,0,0,0] => [(1,10),(2,3),(4,5),(6,9),(7,8)] => [3,5,2,8,4,9,10,7,6,1] => 110101100 => 9
[1,1,0,1,1,0,0,0,1,0] => [(1,8),(2,3),(4,7),(5,6),(9,10)] => [3,6,2,7,8,5,4,1,10,9] => 110110001 => 13
[1,1,0,1,1,0,0,1,0,0] => [(1,10),(2,3),(4,7),(5,6),(8,9)] => [3,6,2,7,9,5,4,10,8,1] => 110110010 => 10
[1,1,0,1,1,0,1,0,0,0] => [(1,10),(2,3),(4,9),(5,6),(7,8)] => [3,6,2,8,9,5,10,7,4,1] => 110110100 => 7
[1,1,0,1,1,1,0,0,0,0] => [(1,10),(2,3),(4,9),(5,8),(6,7)] => [3,7,2,8,9,10,6,5,4,1] => 110111000 => 4
[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] => 111000101 => 14
[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] => 111000110 => 10
[1,1,1,0,0,1,0,0,1,0] => [(1,8),(2,5),(3,4),(6,7),(9,10)] => [4,5,7,3,2,8,6,1,10,9] => 111001001 => 12
[1,1,1,0,0,1,0,1,0,0] => [(1,10),(2,5),(3,4),(6,7),(8,9)] => [4,5,7,3,2,9,6,10,8,1] => 111001010 => 9
[1,1,1,0,0,1,1,0,0,0] => [(1,10),(2,5),(3,4),(6,9),(7,8)] => [4,5,8,3,2,9,10,7,6,1] => 111001100 => 6
[1,1,1,0,1,0,0,0,1,0] => [(1,8),(2,7),(3,4),(5,6),(9,10)] => [4,6,7,3,8,5,2,1,10,9] => 111010001 => 9
[1,1,1,0,1,0,0,1,0,0] => [(1,10),(2,7),(3,4),(5,6),(8,9)] => [4,6,7,3,9,5,2,10,8,1] => 111010010 => 7
[1,1,1,0,1,0,1,0,0,0] => [(1,10),(2,9),(3,4),(5,6),(7,8)] => [4,6,8,3,9,5,10,7,2,1] => 111010100 => 5
[1,1,1,0,1,1,0,0,0,0] => [(1,10),(2,9),(3,4),(5,8),(6,7)] => [4,7,8,3,9,10,6,5,2,1] => 111011000 => 3
[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] => 111100001 => 5
[1,1,1,1,0,0,0,1,0,0] => [(1,10),(2,7),(3,6),(4,5),(8,9)] => [5,6,7,9,4,3,2,10,8,1] => 111100010 => 4
[1,1,1,1,0,0,1,0,0,0] => [(1,10),(2,9),(3,6),(4,5),(7,8)] => [5,6,8,9,4,3,10,7,2,1] => 111100100 => 3
[1,1,1,1,0,1,0,0,0,0] => [(1,10),(2,9),(3,8),(4,5),(6,7)] => [5,7,8,9,4,10,6,3,2,1] => 111101000 => 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] => 111110000 => 1
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Description
The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See St001312Number of parabolic noncrossing partitions indexed by the composition. for this statistic on compositions treated as bounce paths.
Map
descent bottoms
Description
The descent bottoms of a permutation as a binary word.
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.