Identifier
Values
{{1}} => {{1}} => [1] => ([],1) => 1
{{1,2}} => {{1,2}} => [2,1] => ([(0,1)],2) => 2
{{1},{2}} => {{1},{2}} => [1,2] => ([(0,1)],2) => 2
{{1,2,3}} => {{1,2,3}} => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
{{1,2},{3}} => {{1,2},{3}} => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
{{1,3},{2}} => {{1},{2,3}} => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 3
{{1},{2,3}} => {{1,3},{2}} => [3,2,1] => ([(0,2),(2,1)],3) => 3
{{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => ([(0,2),(2,1)],3) => 3
{{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 5
{{1,2,3},{4}} => {{1,2,3},{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) => 6
{{1,2,4},{3}} => {{1,2},{3,4}} => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 4
{{1,2},{3,4}} => {{1,2,4},{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) => 6
{{1,2},{3},{4}} => {{1,2},{3},{4}} => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 5
{{1,3,4},{2}} => {{1},{2,3,4}} => [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) => 6
{{1,3},{2,4}} => {{1,4},{2,3}} => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 4
{{1,3},{2},{4}} => {{1},{2,3},{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) => 6
{{1,4},{2,3}} => {{1,3},{2,4}} => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 4
{{1},{2,3,4}} => {{1,3,4},{2}} => [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) => 6
{{1},{2,3},{4}} => {{1,3},{2},{4}} => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 5
{{1,4},{2},{3}} => {{1},{2},{3,4}} => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 5
{{1},{2,4},{3}} => {{1},{2,4},{3}} => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 5
{{1},{2},{3,4}} => {{1,4},{2},{3}} => [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) => 6
{{1},{2},{3},{4}} => {{1},{2},{3},{4}} => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
{{1},{2,4},{3,5}} => {{1,5},{2,4},{3}} => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1},{2},{3},{4},{5}} => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
{{1,4},{2,5},{3,6}} => {{1,6},{2,5},{3,4}} => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
{{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) => 6
{{1},{2,5},{3,6},{4,7}} => {{1,7},{2,6},{3,5},{4}} => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
{{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) => 7
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Description
The dimension of the reduced incidence algebra of a poset.
The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets.
Thus, this statistic returns the number of non-isomorphic intervals of the poset.
Map
Yip
Description
A transformation of set partitions due to Yip.
Return the set partition of $\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Yip's bijection $\psi$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing 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.