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Matching statistic: St000352
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(load all 13 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000352: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 0
{{1,2}}
=> [2,1] => 1
{{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> [2,3,1] => 1
{{1,2},{3}}
=> [2,1,3] => 1
{{1,3},{2}}
=> [3,2,1] => 1
{{1},{2,3}}
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => 1
{{1,2,4},{3}}
=> [2,4,3,1] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => 1
{{1,3},{2,4}}
=> [3,4,1,2] => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 2
Description
The Elizalde-Pak rank of a permutation.
This is the largest $k$ such that $\pi(i) > k$ for all $i\leq k$.
According to [1], the length of the longest increasing subsequence in a $321$-avoiding permutation is equidistributed with the rank of a $132$-avoiding permutation.
Matching statistic: St000120
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 0
{{1,2}}
=> [2,1] => [1,2] => [1,0,1,0]
=> 1
{{1},{2}}
=> [1,2] => [2,1] => [1,1,0,0]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 0
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 1
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
Description
The number of left tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St000971
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 2 = 1 + 1
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 2 = 1 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,5,1,3,2] => {{1,4},{2,5},{3}}
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => {{1,3},{2,5},{4}}
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 3 = 2 + 1
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St000297
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => => ? = 0
{{1,2}}
=> [2,1] => [2,1] => 1 => 1
{{1},{2}}
=> [1,2] => [1,2] => 0 => 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => 10 => 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 10 => 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => 10 => 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 01 => 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 00 => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 100 => 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 100 => 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 100 => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 101 => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 100 => 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 110 => 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 100 => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 110 => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 010 => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 010 => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 100 => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 010 => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 001 => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => 1000 => 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => 1000 => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => 1010 => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => 1000 => 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => 1010 => 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => 1000 => 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => 1010 => 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => 1000 => 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => 1100 => 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => 1000 => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => 1100 => 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => 1100 => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => 1100 => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => 1000 => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => 1100 => 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => 1001 => 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => 1000 => 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => 1100 => 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => 1100 => 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => 1100 => 2
Description
The number of leading ones in a binary word.
Matching statistic: St000621
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [[1],[]]
=> []
=> ? = 0
{{1,2}}
=> [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1}
{{1,2,3}}
=> [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1}
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1}
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
{{1,2,3,4}}
=> [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2}
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2}
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> [2]
=> 1
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2}
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2}
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2}
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2}
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2}
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2}
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> [3]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> [2]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,5},{2,3,4}}
=> [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4,5}}
=> [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4},{5}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1},{2,3,5},{4}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
{{1,4,5},{2},{3}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1,4},{2},{3,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
{{1},{2,4,5},{3}}
=> [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2},{3,4,5}}
=> [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4,5,6}}
=> [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1,2,3,4,5},{6}}
=> [5,1] => [[5,5],[4]]
=> [4]
=> 1
{{1,2,3,4,6},{5}}
=> [5,1] => [[5,5],[4]]
=> [4]
=> 1
{{1,2,3,4},{5,6}}
=> [4,2] => [[5,4],[3]]
=> [3]
=> 0
{{1,2,3,4},{5},{6}}
=> [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
{{1,2,3,5,6},{4}}
=> [5,1] => [[5,5],[4]]
=> [4]
=> 1
{{1,2,3,5},{4,6}}
=> [4,2] => [[5,4],[3]]
=> [3]
=> 0
{{1,2},{3,4,5,6}}
=> [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1,3},{2,4,5,6}}
=> [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1,4},{2,3,5,6}}
=> [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1,5},{2,3,4,6}}
=> [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1,6},{2,3,4,5}}
=> [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2,3,4,5,6}}
=> [1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2,3},{4,5,6}}
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2,4},{3,5,6}}
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2,5},{3,4,6}}
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2,6},{3,4,5}}
=> [1,2,3] => [[4,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2},{3,4,5,6}}
=> [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2},{3,4},{5,6}}
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2},{3,5},{4,6}}
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2},{3,6},{4,5}}
=> [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2},{3},{4,5,6}}
=> [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2},{3},{4,5},{6}}
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2},{3},{4,6},{5}}
=> [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
{{1},{2},{3},{4},{5,6}}
=> [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,3,3,3,3,3,3}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
Matching statistic: St000260
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 75%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> ? = 0
{{1,2,3}}
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0}
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0}
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,2}
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3,4}}
=> [2,1,4,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,2}
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,2}
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,2}
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2,3,4}}
=> [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,2}
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,2}
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2}
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [5,2,3,1,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [4,2,3,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [4,2,3,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,5},{3,4},{6}}
=> [2,5,4,3,1,6] => [5,4,2,3,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [4,2,1,5,3,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [2,5,3,4,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3},{4,5},{6}}
=> [2,1,3,5,4,6] => [5,2,1,3,4,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [1,3,6,4,5,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [3,2,4,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,3,4},{2,5},{6}}
=> [3,5,4,1,2,6] => [1,5,3,4,2,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [3,2,4,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,3,5},{2,4,6}}
=> [3,4,5,6,1,2] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001632
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 75%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [1] => ([],1)
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => ([],2)
=> ? = 0
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,1}
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => ([],3)
=> ? ∊ {0,0,1}
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1}
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,2}
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,2}
{{1,2},{3,4}}
=> [2,1,4,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,2}
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,2}
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ([],4)
=> ? ∊ {0,0,0,1,1,1,1,2}
{{1},{2,3,4}}
=> [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,2}
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,2}
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,2}
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 0
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2}
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [6,2,3,4,1,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [5,2,3,1,6,4] => ([(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [5,2,3,1,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [6,5,2,3,1,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [4,2,3,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [2,6,1,4,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [4,2,5,6,3,1] => ([(1,5),(2,3),(2,5),(5,4)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [4,2,5,6,1,3] => ([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> 0
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> 2
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [6,4,2,3,5,1] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3}
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5)],6)
=> 0
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [4,2,3,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [2,5,1,4,6,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> 0
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001604
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1]
=> [1]
=> []
=> ? = 0
{{1,2}}
=> [2]
=> [2]
=> []
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1}
{{1,2,3}}
=> [3]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
{{1,2},{3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1}
{{1,3},{2}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1}
{{1},{2,3}}
=> [2,1]
=> [3]
=> []
=> ? ∊ {0,0,1,1,1}
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,1]
=> [1,1]
=> ? ∊ {0,0,1,1,1}
{{1,2,3,4}}
=> [4]
=> [2,2]
=> [2]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4,5},{2,3}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3,5}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,5},{2,3,4}}
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4,5}}
=> [4,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> [2,2,2]
=> [2,2]
=> 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,4,5,6},{3}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,4},{3,5,6}}
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,4},{3,5},{6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,4},{3,6},{5}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,4},{3},{5,6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,5},{3,4,6}}
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2,5},{3,4},{6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,6},{3,4,5}}
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
{{1,2},{3,4,5},{6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,6},{3,4},{5}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2},{3,4,6},{5}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,5},{3,6},{4}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,5},{3},{4,6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,6},{3,5},{4}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2},{3,5,6},{4}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,6},{3},{4,5}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2},{3},{4,5,6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,3,4,5,6},{2}}
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,3,4},{2,5,6}}
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
{{1,3,4},{2,5},{6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,3,4},{2,6},{5}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,3,4},{2},{5,6}}
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,3,5},{2,4,6}}
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
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: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => [[1],[]]
=> ([],1)
=> ? = 0
{{1,2}}
=> [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,1}
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,1}
{{1,2,3}}
=> [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,1,1,1}
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,1,1,1}
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,1,1,1}
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,1,1,1}
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,1,1,1}
{{1,2,3,4}}
=> [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,2,2}
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4,5},{2}}
=> [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,4},{2},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3,5},{2,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4,5},{2,3}}
=> [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,4},{2,3},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,3,4}}
=> [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4,5}}
=> [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3,4},{5}}
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1,5},{2,3},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,3,5},{4}}
=> [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2}
{{1},{2,3},{4,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,4},{2,5,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4,6}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,4,5}}
=> [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000771
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> ? = 1 + 1
{{1,2,3}}
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,1,1} + 1
{{1,3},{2}}
=> [3,2,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,1,1} + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,1,1} + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2} + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2} + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2} + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2} + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2} + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2} + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2} + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2} + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2} + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2} + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [5,2,3,4,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [6,2,3,4,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [4,2,3,1,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [4,2,3,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [5,2,3,1,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [6,2,3,1,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [3,2,1,6,5,4] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [6,2,3,5,1,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [3,2,1,6,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [3,2,6,4,1,5] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [6,2,1,4,5,3] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [6,2,4,3,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [5,2,6,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [6,2,5,3,4,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,5},{3,4,6}}
=> [2,5,4,6,1,3] => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,6},{3,4,5}}
=> [2,6,4,5,3,1] => [6,2,1,5,3,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,2,6},{3,4},{5}}
=> [2,6,4,3,5,1] => [6,2,5,3,1,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
{{1,2,5},{3,6},{4}}
=> [2,5,6,4,1,3] => [3,2,6,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001621The number of atoms of a lattice. St001624The breadth of a lattice.
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