Processing math: 100%

Identifier
Values
{{1}} => [1] => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => [2,1] => 0
{{1},{2}} => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [3,1,2] => [3,1,2] => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}} => [3,2,1] => [2,3,1] => [2,3,1] => 1
{{1},{2,3}} => [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
{{1,2,3},{4}} => [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
{{1,2,4},{3}} => [2,4,3,1] => [3,4,1,2] => [3,4,1,2] => 1
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}} => [3,2,4,1] => [2,4,1,3] => [2,4,1,3] => 1
{{1,3},{2,4}} => [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}} => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 1
{{1},{2,3,4}} => [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 2
{{1},{2,4},{3}} => [1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 1
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,4,5,2,3] => [1,4,5,2,3] => 1
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => 1
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,4,2,5,3] => [1,4,2,5,3] => 1
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 1
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => 2
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 1
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
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Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation πHn, a triple ni<j<kn is an occurrence of the type-B 231 pattern, if 1j<k, π(i)<π(j) and π(i) is one larger than π(k), i.e., π(i)=π(k)+1 if π(k)1 and π(i)=1 otherwise.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
inverse first fundamental transformation
Description
Let σ=(i11i1k1)(i1ik) 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 σ to the permutation [i11,,i1k1,,i1,,ik] 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.
Map
to signed permutation
Description
The signed permutation with all signs positive.