Identifier
-
Mp00037:
Graphs
—to partition of connected components⟶
Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤ
Values
([],1) => [1] => [1,0] => [1] => 1
([],2) => [1,1] => [1,1,0,0] => [2,1] => 2
([(0,1)],2) => [2] => [1,0,1,0] => [1,2] => 0
([],3) => [1,1,1] => [1,1,0,1,0,0] => [2,3,1] => 3
([(1,2)],3) => [2,1] => [1,0,1,1,0,0] => [1,3,2] => 1
([(0,2),(1,2)],3) => [3] => [1,0,1,0,1,0] => [1,2,3] => 0
([(0,1),(0,2),(1,2)],3) => [3] => [1,0,1,0,1,0] => [1,2,3] => 0
([],4) => [1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 4
([(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 2
([(1,3),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
([(0,3),(1,3),(2,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
([(0,3),(1,2)],4) => [2,2] => [1,1,1,0,0,0] => [3,1,2] => 0
([(0,3),(1,2),(2,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
([(1,2),(1,3),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 1
([(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
([(0,2),(0,3),(1,2),(1,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
([],5) => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 5
([(3,4)],5) => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 3
([(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
([(1,4),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
([(0,4),(1,4),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(1,4),(2,3)],5) => [2,2,1] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => 1
([(1,4),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
([(0,1),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => 0
([(2,3),(2,4),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 2
([(0,4),(1,4),(2,3),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,4),(1,3),(2,3),(2,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,1),(2,3),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
([],6) => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 6
([(4,5)],6) => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 4
([(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 3
([(2,5),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
([(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(2,5),(3,4)],6) => [2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => 2
([(2,5),(3,4),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
([(1,2),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => 1
([(3,4),(3,5),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 3
([(1,5),(2,5),(3,4),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,1),(2,5),(3,5),(4,5)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 0
([(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(2,4),(2,5),(3,4),(3,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
([(0,5),(1,5),(2,4),(3,4)],6) => [3,3] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(0,5),(1,4),(2,3)],6) => [2,2,2] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => 0
([(1,5),(2,4),(3,4),(3,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,1),(2,5),(3,4),(4,5)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 0
([(1,2),(3,4),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 0
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Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.
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
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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