Your data matches 34 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000648
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => 0
{{1,2}}
 => [2,1] => 0
{{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => 0
{{1,2},{3}}
 => [2,1,3] => 0
{{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] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => 0
{{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] => 0
{{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] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => 1
{{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] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 0
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => 3
{{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] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 1
{{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] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => 1
Description
The number of 2-excedences of a permutation. This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St000731
(load all 5 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => 0
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => 0
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,5,4] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,4,3,5] => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,4,3] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,4,5,3] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,3,2,4,5] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,5,2,3,4] => 0
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,3,2,5,4] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,4,3,2,5] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,5,3,4,2] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,3,5,2] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,3,4,2,5] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,5,4,3,2] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,3,5,4,2] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,3,4,5,2] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,4,5] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,3,2,4] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,3,5,4] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1,2,3,5] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,1,2,4,3] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,2,5,3] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,1,4,3,5] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,1,4,2,3] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,4,3] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,1,4,5,3] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,4,5] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,2,1,3,4] => 0
Description
The number of double exceedences of a permutation. A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000039
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1,0]
 => [1] => 0
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 3
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [5,2,4,3,1] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => 0
Description
The number of crossings of a permutation. A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$. Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St000366
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => 0
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [3,2,1] => [2,3,1] => 0
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => [1,3,2] => 0
{{1},{2,3}}
 => [1,3,2] => [2,1,3] => [2,1,3] => 0
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => [3,2,1] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,2,4,1] => [2,4,3,1] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,3,1] => [3,4,2,1] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,3,4,2] => [1,4,3,2] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => [4,3,2,1] => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [5,2,3,4,1] => [2,3,4,5,1] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,2,3,1,5] => [2,3,4,1,5] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [4,2,3,5,1] => [2,3,5,4,1] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [5,2,1,3,4] => [2,5,1,3,4] => 0
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [5,2,4,3,1] => [2,4,3,5,1] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,2,1,4,5] => [2,3,1,4,5] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [3,2,5,4,1] => [2,4,5,3,1] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,4,3] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [3,2,1,5,4] => [2,3,1,5,4] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2,4,1,5] => [2,4,3,1,5] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [3,2,4,5,1] => [2,5,4,3,1] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,2,4,3] => [4,5,1,2,3] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [5,3,2,4,1] => [3,2,4,5,1] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,5,2,3,4] => [1,5,2,3,4] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,1,2,3,5] => [4,1,2,3,5] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,5,2,3,1] => [5,2,3,4,1] => 0
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1,2,5,3] => [5,4,1,2,3] => 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [4,3,2,1,5] => [3,2,4,1,5] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [4,3,2,5,1] => [3,2,5,4,1] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,3,2,5] => [1,3,4,2,5] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,1,3,2,4] => [3,5,1,2,4] => 0
Description
The number of double descents of a permutation. A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St001483
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2 = 1 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1 = 0 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000732
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => [1] => ? = 0
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [1,3,2] => [1,3,2] => 0
{{1,3},{2}}
 => [3,2,1] => [2,1,3] => [2,1,3] => 0
{{1},{2,3}}
 => [1,3,2] => [3,2,1] => [3,2,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => [3,1,2] => 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [2,3,1,4] => [3,1,2,4] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 0
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,5,2,3,4] => [1,3,4,5,2] => 0
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 0
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,4,3,5,2] => [1,5,3,2,4] => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,3,4,2,5] => [1,4,2,3,5] => 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,3,5,4,2] => [1,5,2,4,3] => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => 0
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [5,1,3,2,4] => [2,4,3,5,1] => 0
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => 0
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,1,2,3,5] => [2,3,4,1,5] => 0
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [5,1,2,4,3] => [2,3,5,4,1] => 0
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,1,2,5,3] => [2,3,5,1,4] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => 0
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [5,1,4,2,3] => [2,4,5,3,1] => 0
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => 0
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [2,1,4,5,3] => [2,1,5,3,4] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => 0
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [5,2,1,3,4] => [3,2,4,5,1] => 0
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => 0
Description
The number of double deficiencies of a permutation. A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St001176
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 80%
Values
{{1}}
 => [1]
 => []
 => ?
 => ? = 0
{{1,2}}
 => [2]
 => []
 => ?
 => ? ∊ {0,0}
{{1},{2}}
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0}
{{1,2,3}}
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1}
{{1,2},{3}}
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
{{1,3},{2}}
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
{{1},{2,3}}
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
{{1},{2},{3}}
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3,4}}
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,2,3},{4}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,2,4},{3}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,2},{3,4}}
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,2},{3},{4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1,3,4},{2}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,3},{2,4}}
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,3},{2},{4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1,4},{2,3}}
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1},{2,3,4}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1},{2,3},{4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1,4},{2},{3}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2,4},{3}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2},{3,4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2,3,4,5}}
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,3,4},{5}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,3,5},{4}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,3},{4,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,3},{4},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,4},{3,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,4},{3},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2},{3,4,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,5},{3},{4}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3,4,5},{2}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,3,4},{2,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,3,4},{2},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,3},{2,4,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,3,5},{2},{4}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,4,5},{2,3}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,4},{2,3,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1},{2,3,4,5}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1},{2,3,4},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,4,5},{2},{3}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,4,5},{3}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
{{1,2,3,4,5,6}}
 => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,4},{5,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,5},{4,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3},{4,5,6}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,4,5},{3,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,4,6},{3,5}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,4},{3,5,6}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,5,6},{3,4}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,5},{3,4,6}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,6},{3,4,5}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2},{3,4,5,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,4,5},{2,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,4,6},{2,5}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001714
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 80%
Values
{{1}}
 => [1]
 => []
 => ?
 => ? = 0
{{1,2}}
 => [2]
 => []
 => ?
 => ? ∊ {0,0}
{{1},{2}}
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,0}
{{1,2,3}}
 => [3]
 => []
 => ?
 => ? ∊ {0,0,0,1}
{{1,2},{3}}
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
{{1,3},{2}}
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
{{1},{2,3}}
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1}
{{1},{2},{3}}
 => [1,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3,4}}
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,2,3},{4}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,2,4},{3}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,2},{3,4}}
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,2},{3},{4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1,3,4},{2}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,3},{2,4}}
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1,3},{2},{4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1,4},{2,3}}
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1},{2,3,4}}
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
{{1},{2,3},{4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1,4},{2},{3}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2,4},{3}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2},{3,4}}
 => [2,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2,3,4,5}}
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,3,4},{5}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,3,5},{4}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,3},{4,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,3},{4},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,4,5},{3}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,4},{3,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2,4},{3},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,5},{3,4}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2},{3,4,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,2},{3,4},{5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,5},{3},{4}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2},{3,5},{4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2},{3},{4,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3,4,5},{2}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,3,4},{2,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,3,4},{2},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,3,5},{2,4}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,3},{2,4,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,3},{2,4},{5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,3,5},{2},{4}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,3},{2,5},{4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,3},{2},{4,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,4,5},{2,3}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,4},{2,3,5}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1,4},{2,3},{5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,5},{2,3,4}}
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1},{2,3,4,5}}
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3}
{{1},{2,3,4},{5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,5},{2,3},{4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2,3},{4,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,4,5},{2},{3}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1,4},{2,5},{3}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,4},{2},{3,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2,4},{3}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,4,5},{3}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2,4},{3,5}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2},{3,4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2,5},{3,4}}
 => [2,2,1]
 => [2,1]
 => [1]
 => 0
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [1,1]
 => [1]
 => 0
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
{{1,2,3,4,5,6}}
 => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,4},{5,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,4},{5},{6}}
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3,5,6},{4}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,5},{4,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3,5},{4},{6}}
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3,6},{4,5}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3},{4,5,6}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,3},{4,5},{6}}
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,3,6},{4},{5}}
 => [4,1,1]
 => [1,1]
 => [1]
 => 0
{{1,2,3},{4,6},{5}}
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,3},{4},{5,6}}
 => [3,2,1]
 => [2,1]
 => [1]
 => 0
{{1,2,4,5,6},{3}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,4,5},{3,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,4,6},{3,5}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,4},{3,5,6}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,5,6},{3,4}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,5},{3,4,6}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2,6},{3,4,5}}
 => [3,3]
 => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,2},{3,4,5,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,4,5,6},{2}}
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,4,5},{2,6}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
{{1,3,4,6},{2,5}}
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4}
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St000772
(load all 30 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2}}
 => [2] => [1,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1},{2}}
 => [1,1] => [2] => ([],2)
 => ? = 0 + 1
{{1,2,3}}
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,2},{3}}
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,3},{2}}
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1},{2,3}}
 => [1,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0} + 1
{{1},{2},{3}}
 => [1,1,1] => [3] => ([],3)
 => ? ∊ {0,0} + 1
{{1,2,3,4}}
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
{{1,2,3},{4}}
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,2,4},{3}}
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,2},{3},{4}}
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1} + 1
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1} + 1
{{1,4},{2},{3}}
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1} + 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,1} + 1
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => ([],4)
 => ? ∊ {0,0,0,0,1} + 1
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
{{1,2,3,4},{5}}
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,3,5},{4}}
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,3},{4,5}}
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2,4,5},{3}}
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2,5},{3,4}}
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2},{3,4,5}}
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,5},{3},{4}}
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2},{3,5},{4}}
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
{{1,3,4,5},{2}}
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,3},{2,4,5}}
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,3,5},{2},{4}}
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3},{2,5},{4}}
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,4},{2,3,5}}
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,4},{2,3},{5}}
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,5},{2,3,4}}
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,3,4,5}}
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2,3,4},{5}}
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1,5},{2,3},{4}}
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,3,5},{4}}
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2,3},{4,5}}
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1,4,5},{2},{3}}
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4},{2,5},{3}}
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,4},{2},{3,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
{{1,5},{2,4},{3}}
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,4,5},{3}}
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2,4},{3,5}}
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2,4},{3},{5}}
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2,5},{3,4}}
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2},{3,4,5}}
 => [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2,5},{3},{4}}
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2},{3,5},{4}}
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2} + 1
{{1},{2,3,4,5,6}}
 => [1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3,4,5},{6}}
 => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3,4,6},{5}}
 => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3,4},{5,6}}
 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3,4},{5},{6}}
 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3,5,6},{4}}
 => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3,5},{4,6}}
 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3,5},{4},{6}}
 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3,6},{4,5}}
 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3},{4,5,6}}
 => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3},{4,5},{6}}
 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3,6},{4},{5}}
 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3},{4,6},{5}}
 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3},{4},{5,6}}
 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,3},{4},{5},{6}}
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4,5,6},{3}}
 => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4,5},{3,6}}
 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4,5},{3},{6}}
 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4,6},{3,5}}
 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4},{3,5,6}}
 => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4},{3,5},{6}}
 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4,6},{3},{5}}
 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4},{3,6},{5}}
 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4},{3},{5,6}}
 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,4},{3},{5},{6}}
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,5,6},{3,4}}
 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
{{1},{2,5},{3,4,6}}
 => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3} + 1
Description
The multiplicity of the largest 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 $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000771
(load all 26 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
{{1,2}}
 => [2,1] => [1,2] => ([],2)
 => ? = 0 + 1
{{1},{2}}
 => [1,2] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => ([],3)
 => ? ∊ {0,0} + 1
{{1},{2,3}}
 => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,0} + 1
{{1},{2},{3}}
 => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 1 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0} + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [3,5,4,2,1] => ([(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}}
 => [2,5,4,3,1] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2 = 1 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1 = 0 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [5,4,3,1,2] => ([(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] => [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
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [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)
 => 4 = 3 + 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4,5},{2},{3}}
 => [4,2,3,5,1] => [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
{{1,4},{2,5},{3}}
 => [4,5,3,1,2] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,4},{2},{3,5}}
 => [4,2,5,1,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1,4},{2},{3},{5}}
 => [4,2,3,1,5] => [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
{{1,5},{2,4},{3}}
 => [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2,4,5},{3}}
 => [1,4,3,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,4},{3,5}}
 => [1,4,5,2,3] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
{{1},{2,4},{3},{5}}
 => [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,5},{2},{3,4}}
 => [5,2,4,3,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2,5},{3,4}}
 => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
{{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2,5},{3},{4}}
 => [1,5,3,4,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2},{3,5},{4}}
 => [1,2,5,4,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2} + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
{{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 2 + 1
{{1,2,3,4,5},{6}}
 => [2,3,4,5,1,6] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
{{1,2,3,4,6},{5}}
 => [2,3,4,6,5,1] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2,3,4},{5,6}}
 => [2,3,4,1,6,5] => [3,4,5,2,1,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2,3,4},{5},{6}}
 => [2,3,4,1,5,6] => [3,4,5,2,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
{{1,2,3,6},{4,5}}
 => [2,3,6,5,4,1] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2,3},{4,6},{5}}
 => [2,3,1,6,5,4] => [3,4,2,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2,3},{4},{5,6}}
 => [2,3,1,4,6,5] => [3,4,2,5,1,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2,4,6},{3},{5}}
 => [2,4,3,6,5,1] => [3,5,4,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2,4},{3,6},{5}}
 => [2,4,6,1,5,3] => [3,5,1,4,2,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2,4},{3},{5,6}}
 => [2,4,3,1,6,5] => [3,5,4,2,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2},{3,4,6},{5}}
 => [2,1,4,6,5,3] => [3,2,5,1,4,6] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => [3,2,5,4,1,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2,6},{3,5},{4}}
 => [2,6,5,4,3,1] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2,6},{3},{4,5}}
 => [2,6,3,5,4,1] => [3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2},{3,6},{4,5}}
 => [2,1,6,5,4,3] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2},{3,6},{4},{5}}
 => [2,1,6,4,5,3] => [3,2,1,6,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2},{3},{4,6},{5}}
 => [2,1,3,6,5,4] => [3,2,4,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,2},{3},{4},{5,6}}
 => [2,1,3,4,6,5] => [3,2,4,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,3,4,6},{2},{5}}
 => [3,2,4,6,5,1] => [4,3,5,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,3,4},{2},{5,6}}
 => [3,2,4,1,6,5] => [4,3,5,2,1,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,3,6},{2,4,5}}
 => [3,4,6,5,2,1] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,3},{2,4,6},{5}}
 => [3,4,1,6,5,2] => [4,5,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,3},{2,4},{5,6}}
 => [3,4,1,2,6,5] => [4,5,2,3,1,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 1
{{1,3,5},{2,6},{4}}
 => [3,6,5,4,1,2] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3} + 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 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000941The number of characters of the symmetric group whose value on the partition is even. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000260The radius of a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001866The nesting alignments of a signed permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001868The number of alignments of type NE of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation.