Identifier
-
Mp00163:
Signed permutations
—permutation⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤ
Values
[1] => [1] => [1] => ([],1) => 0
[-1] => [1] => [1] => ([],1) => 0
[1,2] => [1,2] => [1,2] => ([(0,1)],2) => 0
[1,-2] => [1,2] => [1,2] => ([(0,1)],2) => 0
[-1,2] => [1,2] => [1,2] => ([(0,1)],2) => 0
[-1,-2] => [1,2] => [1,2] => ([(0,1)],2) => 0
[2,1] => [2,1] => [2,1] => ([(0,1)],2) => 0
[2,-1] => [2,1] => [2,1] => ([(0,1)],2) => 0
[-2,1] => [2,1] => [2,1] => ([(0,1)],2) => 0
[-2,-1] => [2,1] => [2,1] => ([(0,1)],2) => 0
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[-1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[-1,2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[-1,-2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[-1,-2,-3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[1,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,3,-2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,-3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,-3,-2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-1,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-1,3,-2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-1,-3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-1,-3,-2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,1,-3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,-1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,-1,-3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-2,1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-2,1,-3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-2,-1,3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-2,-1,-3] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,3,1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,3,-1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,-3,1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[2,-3,-1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-2,3,1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-2,3,-1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-2,-3,1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-2,-3,-1] => [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[3,1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[3,1,-2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[3,-1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[3,-1,-2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[-3,1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[-3,1,-2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[-3,-1,2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[-3,-1,-2] => [3,1,2] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[3,2,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[3,2,-1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[3,-2,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[3,-2,-1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-3,2,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-3,2,-1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-3,-2,1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[-3,-2,-1] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 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] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 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] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 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] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 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] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 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] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 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] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 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] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 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] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
[4,1,2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[4,1,2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[4,1,-2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[4,1,-2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[4,-1,2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[4,-1,2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[4,-1,-2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[4,-1,-2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[-4,1,2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[-4,1,2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[-4,1,-2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[-4,1,-2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[-4,-1,2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[-4,-1,2,-3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[-4,-1,-2,3] => [4,1,2,3] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[-4,-1,-2,-3] => [4,1,2,3] => [4,3,2,1] => ([(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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [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] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
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Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
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
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.
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.
Map
permutation
Description
The permutation obtained by forgetting the colours.
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