Identifier
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Mp00118:
Dyck paths
—swap returns and last descent⟶
Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤ
Values
[1,0] => [1,0] => [1] => ([],1) => 1
[1,0,1,0] => [1,1,0,0] => [2,1] => ([(0,1)],2) => 1
[1,1,0,0] => [1,0,1,0] => [1,2] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,0,0,1,0] => [1,1,0,1,0,0] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,0,1,0,0] => [1,1,0,0,1,0] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [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
[1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
[1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [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
[1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
[1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [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
[1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => [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
[1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [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
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [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
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [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
[1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [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
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [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
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [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
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => [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
[1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [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
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [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
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
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Description
The number of minimal elements in 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.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
swap returns and last descent
Description
Return a Dyck path with number of returns and length of the last descent interchanged.
This is the specialisation of the map $\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.
This is the specialisation of the map $\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.
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