Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000924
(load all 12 compositions to match this statistic)
Mapping
St000924: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => 1
[(1,2),(3,4)]
 => 2
[(1,3),(2,4)]
 => 1
[(1,4),(2,3)]
 => 2
[(1,2),(3,4),(5,6)]
 => 3
[(1,3),(2,4),(5,6)]
 => 2
[(1,4),(2,3),(5,6)]
 => 3
[(1,5),(2,3),(4,6)]
 => 2
[(1,6),(2,3),(4,5)]
 => 3
[(1,6),(2,4),(3,5)]
 => 2
[(1,5),(2,4),(3,6)]
 => 1
[(1,4),(2,5),(3,6)]
 => 1
[(1,3),(2,5),(4,6)]
 => 1
[(1,2),(3,5),(4,6)]
 => 2
[(1,2),(3,6),(4,5)]
 => 3
[(1,3),(2,6),(4,5)]
 => 2
[(1,4),(2,6),(3,5)]
 => 1
[(1,5),(2,6),(3,4)]
 => 2
[(1,6),(2,5),(3,4)]
 => 3
[(1,2),(3,4),(5,6),(7,8)]
 => 4
[(1,3),(2,4),(5,6),(7,8)]
 => 3
[(1,4),(2,3),(5,6),(7,8)]
 => 4
[(1,5),(2,3),(4,6),(7,8)]
 => 3
[(1,6),(2,3),(4,5),(7,8)]
 => 4
[(1,7),(2,3),(4,5),(6,8)]
 => 3
[(1,8),(2,3),(4,5),(6,7)]
 => 4
[(1,8),(2,4),(3,5),(6,7)]
 => 3
[(1,7),(2,4),(3,5),(6,8)]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => 2
[(1,4),(2,5),(3,6),(7,8)]
 => 2
[(1,3),(2,5),(4,6),(7,8)]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => 3
[(1,2),(3,6),(4,5),(7,8)]
 => 4
[(1,3),(2,6),(4,5),(7,8)]
 => 3
[(1,4),(2,6),(3,5),(7,8)]
 => 2
[(1,5),(2,6),(3,4),(7,8)]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => 4
[(1,7),(2,5),(3,4),(6,8)]
 => 3
[(1,8),(2,5),(3,4),(6,7)]
 => 4
[(1,8),(2,6),(3,4),(5,7)]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => 2
[(1,6),(2,7),(3,4),(5,8)]
 => 2
[(1,5),(2,7),(3,4),(6,8)]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => 1
[(1,3),(2,7),(4,5),(6,8)]
 => 2
[(1,2),(3,7),(4,5),(6,8)]
 => 3
[(1,2),(3,8),(4,5),(6,7)]
 => 4
[(1,3),(2,8),(4,5),(6,7)]
 => 3
[(1,4),(2,8),(3,5),(6,7)]
 => 2
Description
The number of topologically connected components of a perfect matching. For example, the perfect matching $\{\{1,4\},\{2,3\}\}$ has the two connected components $\{1,4\}$ and $\{2,3\}$. The number of perfect matchings with only one block is [[oeis:A000699]].
Matching statistic: St001461
Mapping
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[(1,2)]
 => [2,1] => [2,1] => [1] => 1
[(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 2
[(1,3),(2,4)]
 => [3,4,1,2] => [3,1,4,2] => [3,1,2] => 1
[(1,4),(2,3)]
 => [3,4,2,1] => [4,1,3,2] => [1,3,2] => 2
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,1,4,3,5] => 3
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,1,4,2,6,5] => [3,1,4,2,5] => 2
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,1,3,2,6,5] => [4,1,3,2,5] => 3
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [5,1,3,2,6,4] => [5,1,3,2,4] => 2
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,1,3,2,5,4] => [1,3,2,5,4] => 3
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [6,1,4,2,5,3] => [1,4,2,5,3] => 2
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [5,1,4,2,6,3] => [5,1,4,2,3] => 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,1,5,2,3] => 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [3,1,5,2,4] => 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,5,3,6,4] => [2,1,5,3,4] => 2
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,3,5,4] => [2,1,3,5,4] => 3
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [3,1,6,2,5,4] => [3,1,2,5,4] => 2
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [4,1,6,2,5,3] => [4,1,2,5,3] => 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [6,2,5,1,4,3] => [2,5,1,4,3] => 2
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [5,2,6,1,4,3] => [5,2,1,4,3] => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,7] => ? = 4
[(1,3),(2,4),(5,6),(7,8)]
 => [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => [3,1,4,2,6,5,7] => ? = 3
[(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [4,1,3,2,6,5,8,7] => [4,1,3,2,6,5,7] => ? = 4
[(1,5),(2,3),(4,6),(7,8)]
 => [3,5,2,6,1,4,8,7] => [5,1,3,2,6,4,8,7] => [5,1,3,2,6,4,7] => ? = 3
[(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [6,1,3,2,5,4,8,7] => [6,1,3,2,5,4,7] => ? = 4
[(1,7),(2,3),(4,5),(6,8)]
 => [3,5,2,7,4,8,1,6] => [7,1,3,2,5,4,8,6] => [7,1,3,2,5,4,6] => ? = 3
[(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [8,1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => 4
[(1,8),(2,4),(3,5),(6,7)]
 => [4,5,7,2,3,8,6,1] => [8,1,4,2,5,3,7,6] => [1,4,2,5,3,7,6] => 3
[(1,7),(2,4),(3,5),(6,8)]
 => [4,5,7,2,3,8,1,6] => [7,1,4,2,5,3,8,6] => [7,1,4,2,5,3,6] => ? = 2
[(1,6),(2,4),(3,5),(7,8)]
 => [4,5,6,2,3,1,8,7] => [6,1,4,2,5,3,8,7] => [6,1,4,2,5,3,7] => ? = 3
[(1,5),(2,4),(3,6),(7,8)]
 => [4,5,6,2,1,3,8,7] => [5,1,4,2,6,3,8,7] => [5,1,4,2,6,3,7] => ? = 2
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => [4,1,5,2,6,3,7] => ? = 2
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => [3,1,5,2,6,4,7] => ? = 2
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => [2,1,5,3,6,4,7] => ? = 3
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,6,3,5,4,8,7] => [2,1,6,3,5,4,7] => ? = 4
[(1,3),(2,6),(4,5),(7,8)]
 => [3,5,1,6,4,2,8,7] => [3,1,6,2,5,4,8,7] => [3,1,6,2,5,4,7] => ? = 3
[(1,4),(2,6),(3,5),(7,8)]
 => [4,5,6,1,3,2,8,7] => [4,1,6,2,5,3,8,7] => [4,1,6,2,5,3,7] => ? = 2
[(1,5),(2,6),(3,4),(7,8)]
 => [4,5,6,3,1,2,8,7] => [6,2,5,1,4,3,8,7] => [6,2,5,1,4,3,7] => ? = 3
[(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [5,2,6,1,4,3,8,7] => [5,2,6,1,4,3,7] => ? = 4
[(1,7),(2,5),(3,4),(6,8)]
 => [4,5,7,3,2,8,1,6] => [5,2,7,1,4,3,8,6] => [5,2,7,1,4,3,6] => ? = 3
[(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [5,2,8,1,4,3,7,6] => [5,2,1,4,3,7,6] => ? = 4
[(1,8),(2,6),(3,4),(5,7)]
 => [4,6,7,3,8,2,5,1] => [6,2,8,1,4,3,7,5] => [6,2,1,4,3,7,5] => ? = 3
[(1,7),(2,6),(3,4),(5,8)]
 => [4,6,7,3,8,2,1,5] => [6,2,7,1,4,3,8,5] => [6,2,7,1,4,3,5] => ? = 2
[(1,6),(2,7),(3,4),(5,8)]
 => [4,6,7,3,8,1,2,5] => [7,2,6,1,4,3,8,5] => [7,2,6,1,4,3,5] => ? = 2
[(1,5),(2,7),(3,4),(6,8)]
 => [4,5,7,3,1,8,2,6] => [7,2,5,1,4,3,8,6] => [7,2,5,1,4,3,6] => ? = 2
[(1,4),(2,7),(3,5),(6,8)]
 => [4,5,7,1,3,8,2,6] => [4,1,7,2,5,3,8,6] => [4,1,7,2,5,3,6] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
 => [3,5,1,7,4,8,2,6] => [3,1,7,2,5,4,8,6] => [3,1,7,2,5,4,6] => ? = 2
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,5,7,4,8,3,6] => [2,1,7,3,5,4,8,6] => [2,1,7,3,5,4,6] => ? = 3
[(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [2,1,8,3,5,4,7,6] => [2,1,3,5,4,7,6] => ? = 4
[(1,3),(2,8),(4,5),(6,7)]
 => [3,5,1,7,4,8,6,2] => [3,1,8,2,5,4,7,6] => [3,1,2,5,4,7,6] => ? = 3
[(1,4),(2,8),(3,5),(6,7)]
 => [4,5,7,1,3,8,6,2] => [4,1,8,2,5,3,7,6] => [4,1,2,5,3,7,6] => ? = 2
[(1,5),(2,8),(3,4),(6,7)]
 => [4,5,7,3,1,8,6,2] => [8,2,5,1,4,3,7,6] => [2,5,1,4,3,7,6] => ? = 3
[(1,6),(2,8),(3,4),(5,7)]
 => [4,6,7,3,8,1,5,2] => [8,2,6,1,4,3,7,5] => [2,6,1,4,3,7,5] => ? = 2
[(1,7),(2,8),(3,4),(5,6)]
 => [4,6,7,3,8,5,1,2] => [7,1,4,3,8,2,6,5] => [7,1,4,3,2,6,5] => ? = 3
[(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [8,1,4,3,7,2,6,5] => [1,4,3,7,2,6,5] => 4
[(1,8),(2,7),(3,5),(4,6)]
 => [5,6,7,8,3,4,2,1] => [8,1,5,3,7,2,6,4] => [1,5,3,7,2,6,4] => ? = 3
[(1,7),(2,8),(3,5),(4,6)]
 => [5,6,7,8,3,4,1,2] => [7,1,5,3,8,2,6,4] => [7,1,5,3,2,6,4] => ? = 2
[(1,6),(2,8),(3,5),(4,7)]
 => [5,6,7,8,3,1,4,2] => [8,2,6,1,5,3,7,4] => [2,6,1,5,3,7,4] => ? = 1
[(1,5),(2,8),(3,6),(4,7)]
 => [5,6,7,8,1,3,4,2] => [5,1,8,2,6,3,7,4] => [5,1,2,6,3,7,4] => ? = 1
[(1,4),(2,8),(3,6),(5,7)]
 => [4,6,7,1,8,3,5,2] => [4,1,8,2,6,3,7,5] => [4,1,2,6,3,7,5] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
 => [3,6,1,7,8,4,5,2] => [3,1,8,2,6,4,7,5] => [3,1,2,6,4,7,5] => ? = 2
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,6,7,8,4,5,3] => [2,1,8,3,6,4,7,5] => [2,1,3,6,4,7,5] => ? = 3
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,6,7,8,4,3,5] => [2,1,7,3,6,4,8,5] => [2,1,7,3,6,4,5] => ? = 2
[(1,3),(2,7),(4,6),(5,8)]
 => [3,6,1,7,8,4,2,5] => [3,1,7,2,6,4,8,5] => [3,1,7,2,6,4,5] => ? = 1
[(1,4),(2,7),(3,6),(5,8)]
 => [4,6,7,1,8,3,2,5] => [4,1,7,2,6,3,8,5] => [4,1,7,2,6,3,5] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
 => [5,6,7,8,1,3,2,4] => [5,1,7,2,6,3,8,4] => [5,1,7,2,6,3,4] => ? = 1
[(1,6),(2,7),(3,5),(4,8)]
 => [5,6,7,8,3,1,2,4] => [7,2,6,1,5,3,8,4] => [7,2,6,1,5,3,4] => ? = 1
[(1,7),(2,6),(3,5),(4,8)]
 => [5,6,7,8,3,2,1,4] => [6,2,7,1,5,3,8,4] => [6,2,7,1,5,3,4] => ? = 1
[(1,8),(2,6),(3,5),(4,7)]
 => [5,6,7,8,3,2,4,1] => [6,2,8,1,5,3,7,4] => [6,2,1,5,3,7,4] => ? = 2
[(1,8),(2,5),(3,6),(4,7)]
 => [5,6,7,8,2,3,4,1] => [8,1,5,2,6,3,7,4] => [1,5,2,6,3,7,4] => ? = 2
[(1,7),(2,5),(3,6),(4,8)]
 => [5,6,7,8,2,3,1,4] => [7,1,5,2,6,3,8,4] => [7,1,5,2,6,3,4] => ? = 1
[(1,6),(2,5),(3,7),(4,8)]
 => [5,6,7,8,2,1,3,4] => [6,1,5,2,7,3,8,4] => [6,1,5,2,7,3,4] => ? = 1
[(1,5),(2,6),(3,7),(4,8)]
 => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [5,1,6,2,7,3,4] => ? = 1
[(1,8),(2,4),(3,6),(5,7)]
 => [4,6,7,2,8,3,5,1] => [8,1,4,2,6,3,7,5] => [1,4,2,6,3,7,5] => 2
[(1,8),(2,3),(4,6),(5,7)]
 => [3,6,2,7,8,4,5,1] => [8,1,3,2,6,4,7,5] => [1,3,2,6,4,7,5] => 3
Description
The number of topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$. This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.