Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001491
(load all 2 compositions to match this statistic)
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [1,2] => 1 => 1
[+,3,2] => [1,3,2] => [2,1,3] => 01 => 1
[-,3,2] => [1,3,2] => [2,1,3] => 01 => 1
[3,1,2] => [3,1,2] => [1,3,2] => 10 => 1
[3,+,1] => [3,2,1] => [1,2,3] => 11 => 2
[3,-,1] => [3,2,1] => [1,2,3] => 11 => 2
[+,+,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[-,+,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[+,-,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[-,-,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[+,4,2,3] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[-,4,2,3] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[+,4,+,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[-,4,+,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[+,4,-,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[-,4,-,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 001 => 1
[2,4,+,1] => [2,4,3,1] => [3,1,2,4] => 001 => 1
[2,4,-,1] => [2,4,3,1] => [3,1,2,4] => 001 => 1
[4,1,2,3] => [4,1,2,3] => [1,3,4,2] => 100 => 1
[4,1,+,2] => [4,1,3,2] => [1,3,2,4] => 101 => 2
[4,1,-,2] => [4,1,3,2] => [1,3,2,4] => 101 => 2
[4,+,1,3] => [4,2,1,3] => [1,4,3,2] => 100 => 1
[4,-,1,3] => [4,2,1,3] => [1,4,3,2] => 100 => 1
[4,+,+,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,-,+,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,+,-,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,-,-,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 110 => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 111 => 3
[+,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[-,+,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[+,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[+,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[-,-,+,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[-,+,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[+,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[-,-,-,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0001 => 1
[+,+,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[-,+,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[+,-,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[-,-,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => 0010 => 1
[+,+,5,+,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[-,+,5,+,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[+,-,5,+,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[+,+,5,-,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[-,-,5,+,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[-,+,5,-,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[+,-,5,-,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
[-,-,5,-,3] => [1,2,5,4,3] => [2,3,1,4,5] => 0011 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001603
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Values
[2,1] => [2,1] => [2]
 => []
 => ? = 1
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? = 1
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? = 1
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? = 1
[3,+,1] => [3,2,1] => [3]
 => []
 => ? = 2
[3,-,1] => [3,2,1] => [3]
 => []
 => ? = 2
[+,+,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => ? = 1
[-,+,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => ? = 1
[+,-,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => ? = 1
[-,-,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => ? = 1
[+,4,2,3] => [1,4,2,3] => [2,1,1]
 => [1,1]
 => ? = 1
[-,4,2,3] => [1,4,2,3] => [2,1,1]
 => [1,1]
 => ? = 1
[+,4,+,2] => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1
[-,4,+,2] => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1
[+,4,-,2] => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1
[-,4,-,2] => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => ? = 1
[2,4,+,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? = 1
[2,4,-,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? = 1
[4,1,2,3] => [4,1,2,3] => [2,1,1]
 => [1,1]
 => ? = 1
[4,1,+,2] => [4,1,3,2] => [3,1]
 => [1]
 => ? = 2
[4,1,-,2] => [4,1,3,2] => [3,1]
 => [1]
 => ? = 2
[4,+,1,3] => [4,2,1,3] => [3,1]
 => [1]
 => ? = 1
[4,-,1,3] => [4,2,1,3] => [3,1]
 => [1]
 => ? = 1
[4,+,+,1] => [4,2,3,1] => [3,1]
 => [1]
 => ? = 1
[4,-,+,1] => [4,2,3,1] => [3,1]
 => [1]
 => ? = 1
[4,+,-,1] => [4,2,3,1] => [3,1]
 => [1]
 => ? = 1
[4,-,-,1] => [4,2,3,1] => [3,1]
 => [1]
 => ? = 1
[4,3,1,2] => [4,3,1,2] => [3,1]
 => [1]
 => ? = 1
[4,3,2,1] => [4,3,2,1] => [4]
 => []
 => ? = 3
[+,+,+,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[-,+,+,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[+,-,+,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[+,+,-,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[-,-,+,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[-,+,-,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[+,-,-,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[-,-,-,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[+,+,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[-,+,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[+,-,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[-,-,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1
[+,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1
[-,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 1
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? = 1
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? = 1
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? = 1
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? = 1
[+,5,2,3,4] => [1,5,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[-,5,2,3,4] => [1,5,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => 1
[+,5,2,+,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => ? = 0
[-,5,2,+,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => ? = 0
[+,5,2,-,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => ? = 0
[-,5,2,-,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => ? = 0
[+,5,+,2,4] => [1,5,3,2,4] => [3,1,1]
 => [1,1]
 => ? = 1
[-,5,+,2,4] => [1,5,3,2,4] => [3,1,1]
 => [1,1]
 => ? = 1
[+,5,-,2,4] => [1,5,3,2,4] => [3,1,1]
 => [1,1]
 => ? = 1
[-,5,-,2,4] => [1,5,3,2,4] => [3,1,1]
 => [1,1]
 => ? = 1
[2,1,+,5,4] => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1
[2,1,-,5,4] => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 1
[2,1,5,3,4] => [2,1,5,3,4] => [2,2,1]
 => [2,1]
 => 1
[2,3,1,5,4] => [2,3,1,5,4] => [2,2,1]
 => [2,1]
 => 1
[3,1,2,5,4] => [3,1,2,5,4] => [2,2,1]
 => [2,1]
 => 1
[5,1,2,3,4] => [5,1,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. Two colourings are considered equal, if they are obtained by an action of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001604
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 20%
Values
[2,1] => [2,1] => [2]
 => []
 => ? = 1 - 1
[+,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? = 1 - 1
[-,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? = 1 - 1
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? = 1 - 1
[3,+,1] => [3,2,1] => [3]
 => []
 => ? = 2 - 1
[3,-,1] => [3,2,1] => [3]
 => []
 => ? = 2 - 1
[+,+,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => ? = 1 - 1
[-,+,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => ? = 1 - 1
[+,-,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => ? = 1 - 1
[-,-,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => ? = 1 - 1
[+,4,2,3] => [1,4,2,3] => [2,1,1]
 => [1,1]
 => ? = 1 - 1
[-,4,2,3] => [1,4,2,3] => [2,1,1]
 => [1,1]
 => ? = 1 - 1
[+,4,+,2] => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1 - 1
[-,4,+,2] => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1 - 1
[+,4,-,2] => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1 - 1
[-,4,-,2] => [1,4,3,2] => [3,1]
 => [1]
 => ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => ? = 1 - 1
[2,4,+,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? = 1 - 1
[2,4,-,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? = 1 - 1
[4,1,2,3] => [4,1,2,3] => [2,1,1]
 => [1,1]
 => ? = 1 - 1
[4,1,+,2] => [4,1,3,2] => [3,1]
 => [1]
 => ? = 2 - 1
[4,1,-,2] => [4,1,3,2] => [3,1]
 => [1]
 => ? = 2 - 1
[4,+,1,3] => [4,2,1,3] => [3,1]
 => [1]
 => ? = 1 - 1
[4,-,1,3] => [4,2,1,3] => [3,1]
 => [1]
 => ? = 1 - 1
[4,+,+,1] => [4,2,3,1] => [3,1]
 => [1]
 => ? = 1 - 1
[4,-,+,1] => [4,2,3,1] => [3,1]
 => [1]
 => ? = 1 - 1
[4,+,-,1] => [4,2,3,1] => [3,1]
 => [1]
 => ? = 1 - 1
[4,-,-,1] => [4,2,3,1] => [3,1]
 => [1]
 => ? = 1 - 1
[4,3,1,2] => [4,3,1,2] => [3,1]
 => [1]
 => ? = 1 - 1
[4,3,2,1] => [4,3,2,1] => [4]
 => []
 => ? = 3 - 1
[+,+,+,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-,+,+,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[+,-,+,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[+,+,-,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-,-,+,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-,+,-,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[+,-,-,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-,-,-,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[+,+,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-,+,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[+,-,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-,-,5,3,4] => [1,2,5,3,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[+,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[-,+,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[+,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[+,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[-,-,5,+,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[-,+,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[+,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[-,-,5,-,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[+,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[-,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[+,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[-,3,5,+,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[+,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[-,3,5,-,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[+,5,2,3,4] => [1,5,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[-,5,2,3,4] => [1,5,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
[+,5,2,+,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => ? = 0 - 1
[-,5,2,+,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => ? = 0 - 1
[+,5,2,-,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => ? = 0 - 1
[-,5,2,-,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => ? = 0 - 1
[+,5,+,2,4] => [1,5,3,2,4] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[-,5,+,2,4] => [1,5,3,2,4] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[+,5,-,2,4] => [1,5,3,2,4] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[-,5,-,2,4] => [1,5,3,2,4] => [3,1,1]
 => [1,1]
 => ? = 1 - 1
[2,1,+,5,4] => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[2,1,-,5,4] => [2,1,3,5,4] => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[2,1,5,3,4] => [2,1,5,3,4] => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[2,3,1,5,4] => [2,3,1,5,4] => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[3,1,2,5,4] => [3,1,2,5,4] => [2,2,1]
 => [2,1]
 => 0 = 1 - 1
[5,1,2,3,4] => [5,1,2,3,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000907
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 80%
Values
[2,1] => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[+,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[-,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,+,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,-,1] => [3,2,1] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[+,+,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[-,+,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[+,-,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[-,-,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[+,4,2,3] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[-,4,2,3] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[+,4,+,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[-,4,+,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[+,4,-,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[-,4,-,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[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),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 1 + 1
[2,4,+,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[2,4,-,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[4,1,2,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[4,1,+,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[4,1,-,2] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[4,+,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[4,-,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[4,+,+,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[4,-,+,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[4,+,-,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[4,-,-,1] => [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)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1 + 1
[4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 3 + 1
[+,+,+,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[-,+,+,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[+,-,+,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[+,+,-,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[-,-,+,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[-,+,-,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[+,-,-,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[-,-,-,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[+,+,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1 + 1
[-,+,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1 + 1
[+,-,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1 + 1
[-,-,5,3,4] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1 + 1
[+,+,5,+,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[-,+,5,+,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[+,-,5,+,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[+,+,5,-,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[-,-,5,+,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[-,+,5,-,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[+,-,5,-,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[-,-,5,-,3] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[+,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 1 + 1
[-,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 1 + 1
[+,3,5,+,2] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 1
[-,3,5,+,2] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 1
[+,3,5,-,2] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 1
[-,3,5,-,2] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 1 + 1
[+,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1 + 1
[-,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 1 + 1
[+,5,2,+,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 0 + 1
[-,5,2,+,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 0 + 1
[+,5,2,-,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 0 + 1
[-,5,2,-,3] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 0 + 1
[+,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1 + 1
[-,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1 + 1
[+,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1 + 1
[-,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1 + 1
[+,5,+,+,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1 + 1
[-,5,+,+,2] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1 + 1
[5,4,+,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
[5,4,-,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 4 + 1
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St001624
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 20%
Values
[2,1] => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[+,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
[-,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
[3,1,2] => [3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,+,1] => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,-,1] => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[+,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[-,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[+,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[-,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[+,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[-,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[+,4,+,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[-,4,+,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[+,4,-,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[-,4,-,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2 = 1 + 1
[2,4,+,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[2,4,-,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[4,1,+,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[4,1,-,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[4,+,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 1
[4,-,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 1
[4,+,+,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,-,+,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,+,-,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,-,-,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,3,2,1] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 + 1
[+,+,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[-,+,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[+,-,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[+,+,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[-,-,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[-,+,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[+,-,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[-,-,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[+,+,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[-,+,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[+,-,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[-,-,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[+,+,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[-,+,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[+,-,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[+,+,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[-,-,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[-,+,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[+,-,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[-,-,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[+,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 1 + 1
[-,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 1 + 1
[+,3,5,+,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[-,3,5,+,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[+,3,5,-,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[-,3,5,-,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[+,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 1 + 1
[-,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 1 + 1
[+,5,2,+,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[-,5,2,+,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[+,5,2,-,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[-,5,2,-,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[+,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[-,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[+,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[-,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[+,5,+,+,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 1 + 1
[-,5,+,+,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 1 + 1
[+,5,-,+,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 1 + 1
Description
The breadth of a lattice. The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.
Matching statistic: St001630
Mapping
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 20%
Values
[2,1] => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[+,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
[-,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
[3,1,2] => [3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,+,1] => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[3,-,1] => [3,2,1] => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
[+,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[-,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[+,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[-,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[+,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[-,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[+,4,+,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[-,4,+,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[+,4,-,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[-,4,-,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2 = 1 + 1
[2,4,+,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[2,4,-,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 + 1
[4,1,+,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[4,1,-,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[4,+,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 1
[4,-,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 1
[4,+,+,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,-,+,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,+,-,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,-,-,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,3,2,1] => [4,3,2,1] => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 3 + 1
[+,+,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[-,+,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[+,-,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[+,+,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[-,-,+,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[-,+,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[+,-,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[-,-,-,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[+,+,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[-,+,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[+,-,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[-,-,5,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[+,+,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[-,+,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[+,-,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[+,+,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[-,-,5,+,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[-,+,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[+,-,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[-,-,5,-,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[+,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 1 + 1
[-,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 1 + 1
[+,3,5,+,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[-,3,5,+,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[+,3,5,-,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[-,3,5,-,2] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 1 + 1
[+,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 1 + 1
[-,5,2,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 1 + 1
[+,5,2,+,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[-,5,2,+,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[+,5,2,-,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[-,5,2,-,3] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[+,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[-,5,+,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[+,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[-,5,-,2,4] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[+,5,+,+,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 1 + 1
[-,5,+,+,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 1 + 1
[+,5,-,+,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 1 + 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.