Identifier
Values
{{1}} => [1] => [1] => ([],1) => 0
{{1,2}} => [2,1] => [2,1] => ([(0,1)],2) => 0
{{1},{2}} => [1,2] => [1,2] => ([(0,1)],2) => 0
{{1,2,3}} => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
{{1,3},{2}} => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
{{1},{2,3}} => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
{{1,2,3,4}} => [2,3,4,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
{{1,2,3},{4}} => [2,3,1,4] => [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) => 0
{{1,2,4},{3}} => [2,4,3,1] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 0
{{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) => 0
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
{{1,3},{2},{4}} => [3,2,1,4] => [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) => 0
{{1,4},{2,3}} => [4,3,2,1] => [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) => 0
{{1},{2,3,4}} => [1,3,4,2] => [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) => 0
{{1},{2,3},{4}} => [1,3,2,4] => [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) => 0
{{1,4},{2},{3}} => [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
{{1},{2,4},{3}} => [1,4,3,2] => [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) => 0
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
{{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) => 0
search for individual values
searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
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.
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.