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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000044
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St000044: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 2
[(1,2),(3,4)]
=> 3
[(1,3),(2,4)]
=> 1
[(1,4),(2,3)]
=> 3
[(1,2),(3,4),(5,6)]
=> 4
[(1,3),(2,4),(5,6)]
=> 2
[(1,4),(2,3),(5,6)]
=> 4
[(1,5),(2,3),(4,6)]
=> 2
[(1,6),(2,3),(4,5)]
=> 4
[(1,6),(2,4),(3,5)]
=> 2
[(1,5),(2,4),(3,6)]
=> 2
[(1,4),(2,5),(3,6)]
=> 2
[(1,3),(2,5),(4,6)]
=> 2
[(1,2),(3,5),(4,6)]
=> 2
[(1,2),(3,6),(4,5)]
=> 4
[(1,3),(2,6),(4,5)]
=> 2
[(1,4),(2,6),(3,5)]
=> 2
[(1,5),(2,6),(3,4)]
=> 2
[(1,6),(2,5),(3,4)]
=> 4
[(1,2),(3,4),(5,6),(7,8)]
=> 5
[(1,3),(2,4),(5,6),(7,8)]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> 5
[(1,5),(2,3),(4,6),(7,8)]
=> 3
[(1,6),(2,3),(4,5),(7,8)]
=> 5
[(1,7),(2,3),(4,5),(6,8)]
=> 3
[(1,8),(2,3),(4,5),(6,7)]
=> 5
[(1,8),(2,4),(3,5),(6,7)]
=> 3
[(1,7),(2,4),(3,5),(6,8)]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> 3
[(1,2),(3,5),(4,6),(7,8)]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> 5
[(1,3),(2,6),(4,5),(7,8)]
=> 3
[(1,4),(2,6),(3,5),(7,8)]
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> 5
[(1,7),(2,5),(3,4),(6,8)]
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> 5
[(1,8),(2,6),(3,4),(5,7)]
=> 3
[(1,7),(2,6),(3,4),(5,8)]
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> 3
[(1,5),(2,7),(3,4),(6,8)]
=> 3
[(1,4),(2,7),(3,5),(6,8)]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> 3
[(1,2),(3,7),(4,5),(6,8)]
=> 3
[(1,2),(3,8),(4,5),(6,7)]
=> 5
[(1,3),(2,8),(4,5),(6,7)]
=> 3
[(1,4),(2,8),(3,5),(6,7)]
=> 3
Description
The number of vertices of the unicellular map given by a perfect matching.
If the perfect matching of $2n$ elements is viewed as a fixed point-free involution $\epsilon$ This statistic is counting the number of cycles of the permutation $\gamma \circ \epsilon$ where $\gamma$ is the long cycle $(1,2,3,\ldots,2n)$.
'''Example'''
The perfect matching $[(1,3),(2,4)]$ corresponds to the permutation in $S_4$ with disjoint cycle decomposition $(1,3)(2,4)$. Then the permutation $(1,2,3,4)\circ (1,3)(2,4) = (1,4,3,2)$ has only one cycle.
Let $\epsilon_v(n)$ is the number of matchings of $2n$ such that yield $v$ cycles in the process described above. Harer and Zagier [1] gave the following expression for the generating series of the numbers $\epsilon_v(n)$.
$$
\sum_{v=1}^{n+1} \epsilon_{v}(n) N^v = (2n-1)!! \sum_{k\geq 0}^n \binom{N}{k+1}\binom{n}{k}2^k.
$$
Matching statistic: St000031
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [1,2] => 2
[(1,2),(3,4)]
=> [2,1,4,3] => [3,2,1,4] => 3
[(1,3),(2,4)]
=> [3,4,1,2] => [4,1,2,3] => 1
[(1,4),(2,3)]
=> [4,3,2,1] => [1,4,3,2] => 3
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,2,5,4,1,6] => 4
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,5,2,3,1,6] => 2
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,4,3,2,1,6] => 4
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [6,4,3,1,2,5] => 2
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,4,3,6,5,2] => 4
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [1,5,6,3,4,2] => 2
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [6,5,1,3,2,4] => 2
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [5,6,1,2,3,4] => 2
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [4,6,2,1,3,5] => 2
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,2,6,1,4,5] => 2
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [3,2,1,6,5,4] => 4
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [4,1,2,6,5,3] => 2
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [5,1,6,2,4,3] => 2
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [6,1,5,4,2,3] => 2
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,6,5,4,3,2] => 4
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => 5
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,5,2,3,7,6,1,8] => 3
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [5,4,3,2,7,6,1,8] => 5
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [6,4,3,7,2,5,1,8] => 3
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,4,3,6,5,2,1,8] => 5
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [8,4,3,6,5,1,2,7] => 3
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,4,3,6,5,8,7,2] => 5
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [1,5,6,3,4,8,7,2] => 3
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [8,5,6,3,4,1,2,7] => 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [7,5,6,3,4,2,1,8] => 3
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [6,5,7,3,2,4,1,8] => 3
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [5,6,7,2,3,4,1,8] => 3
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,6,2,7,3,5,1,8] => 3
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [3,2,6,7,4,5,1,8] => 3
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [3,2,7,6,5,4,1,8] => 5
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [4,7,2,6,5,3,1,8] => 3
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [5,7,6,2,4,3,1,8] => 3
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [6,7,5,4,2,3,1,8] => 3
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => 5
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [8,6,5,4,3,1,2,7] => 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,6,5,4,3,8,7,2] => 5
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [1,7,5,4,8,3,6,2] => 3
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [8,7,5,4,1,3,2,6] => 3
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [7,8,5,4,1,2,3,6] => 3
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [6,8,5,4,2,1,3,7] => 3
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [5,8,6,2,4,1,3,7] => 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [4,8,2,6,5,1,3,7] => 3
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [3,2,8,6,5,1,4,7] => 3
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,2,1,6,5,8,7,4] => 5
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [4,1,2,6,5,8,7,3] => 3
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [5,1,6,2,4,8,7,3] => 3
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000010
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> [2,1] => [1,2] => [1,1]
=> 2
[(1,2),(3,4)]
=> [2,1,4,3] => [3,2,1,4] => [2,1,1]
=> 3
[(1,3),(2,4)]
=> [3,4,1,2] => [4,1,2,3] => [4]
=> 1
[(1,4),(2,3)]
=> [4,3,2,1] => [1,4,3,2] => [2,1,1]
=> 3
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,2,5,4,1,6] => [3,1,1,1]
=> 4
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,5,2,3,1,6] => [5,1]
=> 2
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,4,3,2,1,6] => [2,2,1,1]
=> 4
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [6,4,3,1,2,5] => [5,1]
=> 2
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,4,3,6,5,2] => [3,1,1,1]
=> 4
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [1,5,6,3,4,2] => [5,1]
=> 2
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [6,5,1,3,2,4] => [4,2]
=> 2
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [5,6,1,2,3,4] => [3,3]
=> 2
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [4,6,2,1,3,5] => [4,2]
=> 2
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,2,6,1,4,5] => [5,1]
=> 2
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [3,2,1,6,5,4] => [2,2,1,1]
=> 4
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [4,1,2,6,5,3] => [5,1]
=> 2
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [5,1,6,2,4,3] => [4,2]
=> 2
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [6,1,5,4,2,3] => [5,1]
=> 2
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,6,5,4,3,2] => [2,2,1,1]
=> 4
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => [4,1,1,1,1]
=> 5
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,5,2,3,7,6,1,8] => [6,1,1]
=> 3
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [5,4,3,2,7,6,1,8] => [3,2,1,1,1]
=> 5
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [6,4,3,7,2,5,1,8] => [6,1,1]
=> 3
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,4,3,6,5,2,1,8] => [3,2,1,1,1]
=> 5
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [8,4,3,6,5,1,2,7] => [6,1,1]
=> 3
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,4,3,6,5,8,7,2] => [4,1,1,1,1]
=> 5
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [1,5,6,3,4,8,7,2] => [6,1,1]
=> 3
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [8,5,6,3,4,1,2,7] => [8]
=> 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [7,5,6,3,4,2,1,8] => [5,2,1]
=> 3
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [6,5,7,3,2,4,1,8] => [5,2,1]
=> 3
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [5,6,7,2,3,4,1,8] => [4,3,1]
=> 3
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,6,2,7,3,5,1,8] => [4,3,1]
=> 3
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [3,2,6,7,4,5,1,8] => [6,1,1]
=> 3
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [3,2,7,6,5,4,1,8] => [3,2,1,1,1]
=> 5
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [4,7,2,6,5,3,1,8] => [6,1,1]
=> 3
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [5,7,6,2,4,3,1,8] => [5,2,1]
=> 3
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [6,7,5,4,2,3,1,8] => [6,1,1]
=> 3
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => [2,2,2,1,1]
=> 5
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [8,6,5,4,3,1,2,7] => [5,2,1]
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,6,5,4,3,8,7,2] => [3,2,1,1,1]
=> 5
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [1,7,5,4,8,3,6,2] => [6,1,1]
=> 3
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [8,7,5,4,1,3,2,6] => [5,2,1]
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [7,8,5,4,1,2,3,6] => [4,3,1]
=> 3
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [6,8,5,4,2,1,3,7] => [5,2,1]
=> 3
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [5,8,6,2,4,1,3,7] => [8]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [4,8,2,6,5,1,3,7] => [4,3,1]
=> 3
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [3,2,8,6,5,1,4,7] => [6,1,1]
=> 3
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,2,1,6,5,8,7,4] => [3,2,1,1,1]
=> 5
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [4,1,2,6,5,8,7,3] => [6,1,1]
=> 3
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [5,1,6,2,4,8,7,3] => [4,3,1]
=> 3
Description
The length of the partition.
Matching statistic: St000105
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 80%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 80%
Values
[(1,2)]
=> [2,1] => [1,2] => {{1},{2}}
=> 2
[(1,2),(3,4)]
=> [2,1,4,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[(1,3),(2,4)]
=> [3,4,1,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [1,4,3,6,5,2] => {{1},{2,4,6},{3},{5}}
=> 4
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,1,2,6,5,3] => {{1,2,3,4,6},{5}}
=> 2
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,1,6,5,4] => {{1,3},{2},{4,6},{5}}
=> 4
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,6,1,4,5] => {{1,3,4,5,6},{2}}
=> 2
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,1,6] => {{1,3,5},{2},{4},{6}}
=> 4
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,5,2,3,1,6] => {{1,2,3,4,5},{6}}
=> 2
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,6,2,1,3,5] => {{1,4},{2,3,5,6}}
=> 2
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
=> 2
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [5,1,6,2,4,3] => {{1,2,4,5},{3,6}}
=> 2
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [1,5,6,3,4,2] => {{1},{2,3,4,5,6}}
=> 2
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [1,6,5,4,3,2] => {{1},{2,6},{3,5},{4}}
=> 4
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [6,1,5,4,2,3] => {{1,2,3,5,6},{4}}
=> 2
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [6,5,1,3,2,4] => {{1,3,4,6},{2,5}}
=> 2
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [6,4,3,1,2,5] => {{1,2,4,5,6},{3}}
=> 2
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [5,4,3,2,1,6] => {{1,5},{2,4},{3},{6}}
=> 4
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [1,4,3,6,5,8,7,2] => {{1},{2,4,6,8},{3},{5},{7}}
=> ? = 5
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,1,2,6,5,8,7,3] => {{1,2,3,4,6,8},{5},{7}}
=> ? = 3
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,2,1,6,5,8,7,4] => {{1,3},{2},{4,6,8},{5},{7}}
=> ? = 5
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [3,2,6,1,4,8,7,5] => {{1,3,4,5,6,8},{2},{7}}
=> ? = 3
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,1,8,7,6] => {{1,3,5},{2},{4},{6,8},{7}}
=> ? = 5
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,8,1,6,7] => {{1,3,5,6,7,8},{2},{4}}
=> ? = 3
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,1,8] => {{1,3,5,7},{2},{4},{6},{8}}
=> ? = 5
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,5,2,3,7,6,1,8] => {{1,2,3,4,5,7},{6},{8}}
=> ? = 3
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,5,2,3,8,1,6,7] => {{1,2,3,4,5,6,7,8}}
=> ? = 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,5,2,3,1,8,7,6] => {{1,2,3,4,5},{6,8},{7}}
=> ? = 3
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,6,2,1,3,8,7,5] => {{1,4},{2,3,5,6,8},{7}}
=> ? = 3
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [5,6,1,2,3,8,7,4] => {{1,3,5},{2,4,6,8},{7}}
=> ? = 3
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [5,1,6,2,4,8,7,3] => {{1,2,4,5},{3,6,8},{7}}
=> ? = 3
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [1,5,6,3,4,8,7,2] => {{1},{2,3,4,5,6,8},{7}}
=> ? = 3
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [1,6,5,4,3,8,7,2] => {{1},{2,6,8},{3,5},{4},{7}}
=> ? = 5
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [6,1,5,4,2,8,7,3] => {{1,2,3,5,6,8},{4},{7}}
=> ? = 3
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [6,5,1,3,2,8,7,4] => {{1,3,4,6,8},{2,5},{7}}
=> ? = 3
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [6,4,3,1,2,8,7,5] => {{1,2,4,5,6,8},{3},{7}}
=> ? = 3
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [5,4,3,2,1,8,7,6] => {{1,5},{2,4},{3},{6,8},{7}}
=> ? = 5
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [5,4,3,2,8,1,6,7] => {{1,5,6,7,8},{2,4},{3}}
=> ? = 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [5,4,3,2,7,6,1,8] => {{1,5,7},{2,4},{3},{6},{8}}
=> ? = 5
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [6,4,3,7,2,5,1,8] => {{1,2,4,5,6,7},{3},{8}}
=> ? = 3
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [6,4,3,8,2,1,5,7] => {{1,6},{2,4,5,7,8},{3}}
=> ? = 3
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [7,4,3,8,1,2,5,6] => {{1,5,7},{2,4,6,8},{3}}
=> ? = 3
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [7,4,3,1,8,2,6,5] => {{1,2,4,6,7},{3},{5,8}}
=> ? = 3
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [7,5,1,3,8,2,6,4] => {{1,2,3,4,5,6,7,8}}
=> ? = 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [7,1,5,4,8,2,6,3] => {{1,2,6,7},{3,5,8},{4}}
=> ? = 3
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [1,7,5,4,8,3,6,2] => {{1},{2,3,5,6,7,8},{4}}
=> ? = 3
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [1,8,5,4,7,6,3,2] => {{1},{2,8},{3,5,7},{4},{6}}
=> ? = 5
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [8,1,5,4,7,6,2,3] => {{1,2,3,5,7,8},{4},{6}}
=> ? = 3
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [8,5,1,3,7,6,2,4] => {{1,3,4,8},{2,5,7},{6}}
=> ? = 3
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [8,4,3,1,7,6,2,5] => {{1,2,4,5,7,8},{3},{6}}
=> ? = 3
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [8,4,3,7,1,5,2,6] => {{1,5,6,8},{2,4,7},{3}}
=> ? = 3
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [8,4,3,6,5,1,2,7] => {{1,2,4,6,7,8},{3},{5}}
=> ? = 3
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [7,4,3,6,5,2,1,8] => {{1,7},{2,4,6},{3},{5},{8}}
=> ? = 5
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1,8] => {{1,7},{2,3,4,5,6},{8}}
=> ? = 3
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [8,5,6,3,4,1,2,7] => {{1,2,3,4,5,6,7,8}}
=> ? = 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [8,5,7,3,1,4,2,6] => {{1,2,3,4,5,6,7,8}}
=> ? = 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [8,6,7,1,3,4,2,5] => {{1,2,3,4,5,6,7,8}}
=> ? = 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [8,6,1,7,3,5,2,4] => {{1,2,3,4,5,6,7,8}}
=> ? = 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [8,1,6,7,4,5,2,3] => {{1,2,3,4,5,6,7,8}}
=> ? = 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [1,8,6,7,4,5,3,2] => {{1},{2,8},{3,4,5,6,7}}
=> ? = 3
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [1,7,6,8,4,3,5,2] => {{1},{2,4,5,7,8},{3,6}}
=> ? = 3
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [7,1,6,8,4,2,5,3] => {{1,2,3,4,5,6,7,8}}
=> ? = 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [7,6,1,8,3,2,5,4] => {{1,3,5,7},{2,6},{4,8}}
=> ? = 3
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [7,6,8,1,3,2,4,5] => {{1,4,7},{2,6},{3,5,8}}
=> ? = 3
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [7,5,8,3,1,2,4,6] => {{1,2,3,4,5,6,7,8}}
=> ? = 1
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [6,5,8,3,2,1,4,7] => {{1,6},{2,5},{3,4,7,8}}
=> ? = 3
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1,8] => {{1,3,4,6,7},{2,5},{8}}
=> ? = 3
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1,8] => {{1,3,5,7},{2,4,6},{8}}
=> ? = 3
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000314
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 80%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 80%
Values
[(1,2)]
=> [2,1] => [1,2] => [1,2] => 2
[(1,2),(3,4)]
=> [2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 3
[(1,3),(2,4)]
=> [3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 1
[(1,4),(2,3)]
=> [4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 3
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,2,5,4,1,6] => [2,4,5,1,3,6] => 4
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,5,2,3,1,6] => [5,1,4,3,2,6] => 2
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,4,3,2,1,6] => [3,4,2,5,1,6] => 4
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [6,4,3,1,2,5] => [3,6,5,2,4,1] => 2
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,4,3,6,5,2] => [1,3,5,6,2,4] => 4
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [1,5,6,3,4,2] => [1,6,2,5,4,3] => 2
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [6,5,1,3,2,4] => [5,2,6,4,3,1] => 2
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => 2
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [4,6,2,1,3,5] => [4,1,6,5,3,2] => 2
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [3,2,6,1,4,5] => [2,6,5,4,1,3] => 2
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [3,2,1,6,5,4] => [2,3,1,5,6,4] => 4
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [4,1,2,6,5,3] => [5,6,3,2,1,4] => 2
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [5,1,6,2,4,3] => [5,4,2,1,6,3] => 2
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [6,1,5,4,2,3] => [4,6,3,5,2,1] => 2
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => 4
[(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => [2,4,6,7,1,3,5,8] => ? = 5
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,5,2,3,7,6,1,8] => [6,7,1,4,3,2,5,8] => ? = 3
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [5,4,3,2,7,6,1,8] => [3,4,2,6,7,1,5,8] => ? = 5
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [6,4,3,7,2,5,1,8] => [3,7,1,6,5,2,4,8] => ? = 3
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,4,3,6,5,2,1,8] => [3,5,6,2,4,7,1,8] => ? = 5
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [8,4,3,6,5,1,2,7] => [3,5,8,7,2,4,6,1] => ? = 3
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,4,3,6,5,8,7,2] => [1,3,5,7,8,2,4,6] => ? = 5
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [1,5,6,3,4,8,7,2] => [1,7,8,2,5,4,3,6] => ? = 3
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [8,5,6,3,4,1,2,7] => [8,7,2,5,4,3,6,1] => ? = 1
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [7,5,6,3,4,2,1,8] => [6,2,5,4,3,7,1,8] => ? = 3
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [6,5,7,3,2,4,1,8] => [5,2,7,1,6,4,3,8] => ? = 3
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [5,6,7,2,3,4,1,8] => [6,4,2,7,1,5,3,8] => ? = 3
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [4,6,2,7,3,5,1,8] => [6,5,3,2,7,1,4,8] => ? = 3
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [3,2,6,7,4,5,1,8] => [2,7,1,3,6,5,4,8] => ? = 3
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [3,2,7,6,5,4,1,8] => [2,5,6,4,7,1,3,8] => ? = 5
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [4,7,2,6,5,3,1,8] => [5,7,1,4,6,3,2,8] => ? = 3
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [5,7,6,2,4,3,1,8] => [6,3,7,1,5,4,2,8] => ? = 3
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [6,7,5,4,2,3,1,8] => [4,7,1,6,3,5,2,8] => ? = 3
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => [4,5,3,6,2,7,1,8] => ? = 5
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [8,6,5,4,3,1,2,7] => [4,5,3,8,7,2,6,1] => ? = 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,6,5,4,3,8,7,2] => [1,4,5,3,7,8,2,6] => ? = 5
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [1,7,5,4,8,3,6,2] => [1,4,8,2,7,6,3,5] => ? = 3
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [8,7,5,4,1,3,2,6] => [4,7,2,8,6,3,5,1] => ? = 3
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [7,8,5,4,1,2,3,6] => [4,7,3,5,1,8,6,2] => ? = 3
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [6,8,5,4,2,1,3,7] => [4,6,1,8,7,3,5,2] => ? = 3
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [5,8,6,2,4,1,3,7] => [8,7,3,6,1,5,4,2] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [4,8,2,6,5,1,3,7] => [5,6,1,4,8,7,3,2] => ? = 3
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [3,2,8,6,5,1,4,7] => [2,5,8,7,4,6,1,3] => ? = 3
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,2,1,6,5,8,7,4] => [2,3,1,5,7,8,4,6] => ? = 5
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [4,1,2,6,5,8,7,3] => [5,7,8,3,2,1,4,6] => ? = 3
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [5,1,6,2,4,8,7,3] => [5,4,2,1,7,8,3,6] => ? = 3
[(1,5),(2,8),(3,4),(6,7)]
=> [5,8,4,3,1,7,6,2] => [6,1,5,4,2,8,7,3] => [4,7,8,3,5,2,1,6] => ? = 3
[(1,6),(2,8),(3,4),(5,7)]
=> [6,8,4,3,7,1,5,2] => [7,1,5,4,8,2,6,3] => [4,7,6,2,1,8,3,5] => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
=> [7,8,4,3,6,5,1,2] => [8,1,5,4,7,6,2,3] => [4,6,8,3,5,7,2,1] => ? = 3
[(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,8,5,4,7,6,3,2] => [1,4,6,7,3,5,8,2] => ? = 5
[(1,8),(2,7),(3,5),(4,6)]
=> [8,7,5,6,3,4,2,1] => [1,8,6,7,4,5,3,2] => [1,7,3,6,5,4,8,2] => ? = 3
[(1,7),(2,8),(3,5),(4,6)]
=> [7,8,5,6,3,4,1,2] => [8,1,6,7,4,5,2,3] => [8,3,6,5,4,7,2,1] => ? = 1
[(1,6),(2,8),(3,5),(4,7)]
=> [6,8,5,7,3,1,4,2] => [7,1,6,8,4,2,5,3] => [8,3,6,2,1,7,5,4] => ? = 1
[(1,5),(2,8),(3,6),(4,7)]
=> [5,8,6,7,1,3,4,2] => [6,1,7,8,2,4,5,3] => [8,3,7,5,2,1,6,4] => ? = 1
[(1,4),(2,8),(3,6),(5,7)]
=> [4,8,6,1,7,3,5,2] => [5,1,7,2,8,4,6,3] => [8,3,7,6,4,2,1,5] => ? = 1
[(1,3),(2,8),(4,6),(5,7)]
=> [3,8,1,6,7,4,5,2] => [4,1,2,7,8,5,6,3] => [8,3,2,1,4,7,6,5] => ? = 1
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,8,6,7,4,5,3] => [3,2,1,7,8,5,6,4] => [2,3,1,8,4,7,6,5] => ? = 3
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,7,6,8,4,3,5] => [3,2,8,7,1,5,4,6] => [2,7,4,8,6,5,1,3] => ? = 3
[(1,3),(2,7),(4,6),(5,8)]
=> [3,7,1,6,8,4,2,5] => [4,8,2,7,1,5,3,6] => [8,6,5,1,4,7,3,2] => ? = 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,7,6,1,8,3,2,5] => [5,8,7,2,1,4,3,6] => [5,1,7,3,8,6,4,2] => ? = 3
[(1,5),(2,7),(3,6),(4,8)]
=> [5,7,6,8,1,3,2,4] => [6,8,7,1,2,4,3,5] => [6,4,1,7,3,8,5,2] => ? = 3
[(1,6),(2,7),(3,5),(4,8)]
=> [6,7,5,8,3,1,2,4] => [7,8,6,1,4,2,3,5] => [8,5,4,1,7,3,6,2] => ? = 1
[(1,7),(2,6),(3,5),(4,8)]
=> [7,6,5,8,3,2,1,4] => [8,7,6,1,4,3,2,5] => [6,3,7,2,8,5,4,1] => ? = 3
[(1,8),(2,6),(3,5),(4,7)]
=> [8,6,5,7,3,2,4,1] => [1,7,6,8,4,3,5,2] => [1,6,3,8,2,7,5,4] => ? = 3
[(1,8),(2,5),(3,6),(4,7)]
=> [8,5,6,7,2,3,4,1] => [1,6,7,8,3,4,5,2] => [1,7,5,3,8,2,6,4] => ? = 3
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
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