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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000041
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000041: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
=> 0
[(1,2),(3,4)]
=> 0
[(1,3),(2,4)]
=> 0
[(1,4),(2,3)]
=> 1
[(1,2),(3,4),(5,6)]
=> 0
[(1,3),(2,4),(5,6)]
=> 0
[(1,4),(2,3),(5,6)]
=> 1
[(1,5),(2,3),(4,6)]
=> 1
[(1,6),(2,3),(4,5)]
=> 2
[(1,6),(2,4),(3,5)]
=> 2
[(1,5),(2,4),(3,6)]
=> 1
[(1,4),(2,5),(3,6)]
=> 0
[(1,3),(2,5),(4,6)]
=> 0
[(1,2),(3,5),(4,6)]
=> 0
[(1,2),(3,6),(4,5)]
=> 1
[(1,3),(2,6),(4,5)]
=> 1
[(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)]
=> 0
[(1,3),(2,4),(5,6),(7,8)]
=> 0
[(1,4),(2,3),(5,6),(7,8)]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> 3
[(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)]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> 3
[(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)]
=> 4
[(1,7),(2,6),(3,4),(5,8)]
=> 3
[(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)]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> 2
Description
The number of nestings of a perfect matching.
This is the number of pairs of edges ((a,b),(c,d)) such that a≤c≤d≤b. i.e., the edge (c,d) is nested inside (a,b).
Matching statistic: St000042
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St000042: Perfect matchings ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
St000042: Perfect matchings ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
Values
[(1,2)]
=> [(1,2)]
=> 0
[(1,2),(3,4)]
=> [(1,2),(3,4)]
=> 0
[(1,3),(2,4)]
=> [(1,4),(2,3)]
=> 0
[(1,4),(2,3)]
=> [(1,3),(2,4)]
=> 1
[(1,2),(3,4),(5,6)]
=> [(1,2),(3,4),(5,6)]
=> 0
[(1,3),(2,4),(5,6)]
=> [(1,4),(2,3),(5,6)]
=> 0
[(1,4),(2,3),(5,6)]
=> [(1,3),(2,4),(5,6)]
=> 1
[(1,5),(2,3),(4,6)]
=> [(1,3),(2,6),(4,5)]
=> 1
[(1,6),(2,3),(4,5)]
=> [(1,3),(2,5),(4,6)]
=> 2
[(1,6),(2,4),(3,5)]
=> [(1,5),(2,4),(3,6)]
=> 2
[(1,5),(2,4),(3,6)]
=> [(1,6),(2,4),(3,5)]
=> 1
[(1,4),(2,5),(3,6)]
=> [(1,6),(2,5),(3,4)]
=> 0
[(1,3),(2,5),(4,6)]
=> [(1,6),(2,3),(4,5)]
=> 0
[(1,2),(3,5),(4,6)]
=> [(1,2),(3,6),(4,5)]
=> 0
[(1,2),(3,6),(4,5)]
=> [(1,2),(3,5),(4,6)]
=> 1
[(1,3),(2,6),(4,5)]
=> [(1,5),(2,3),(4,6)]
=> 1
[(1,4),(2,6),(3,5)]
=> [(1,5),(2,6),(3,4)]
=> 1
[(1,5),(2,6),(3,4)]
=> [(1,4),(2,6),(3,5)]
=> 2
[(1,6),(2,5),(3,4)]
=> [(1,4),(2,5),(3,6)]
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 0
[(1,3),(2,4),(5,6),(7,8)]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 0
[(1,4),(2,3),(5,6),(7,8)]
=> [(1,3),(2,4),(5,6),(7,8)]
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> [(1,3),(2,6),(4,5),(7,8)]
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> [(1,3),(2,5),(4,6),(7,8)]
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> [(1,3),(2,5),(4,8),(6,7)]
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> [(1,3),(2,5),(4,7),(6,8)]
=> 3
[(1,8),(2,4),(3,5),(6,7)]
=> [(1,5),(2,4),(3,7),(6,8)]
=> 3
[(1,7),(2,4),(3,5),(6,8)]
=> [(1,5),(2,4),(3,8),(6,7)]
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> [(1,5),(2,4),(3,6),(7,8)]
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> [(1,6),(2,4),(3,5),(7,8)]
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> [(1,6),(2,3),(4,5),(7,8)]
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> [(1,2),(3,5),(4,6),(7,8)]
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> [(1,5),(2,3),(4,6),(7,8)]
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> [(1,5),(2,6),(3,4),(7,8)]
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> [(1,4),(2,6),(3,5),(7,8)]
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> [(1,4),(2,5),(3,6),(7,8)]
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> [(1,4),(2,5),(3,8),(6,7)]
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> [(1,4),(2,5),(3,7),(6,8)]
=> 4
[(1,8),(2,6),(3,4),(5,7)]
=> [(1,4),(2,7),(3,6),(5,8)]
=> 4
[(1,7),(2,6),(3,4),(5,8)]
=> [(1,4),(2,8),(3,6),(5,7)]
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> [(1,4),(2,8),(3,7),(5,6)]
=> 2
[(1,5),(2,7),(3,4),(6,8)]
=> [(1,4),(2,8),(3,5),(6,7)]
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> [(1,5),(2,8),(3,4),(6,7)]
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> [(1,5),(2,3),(4,8),(6,7)]
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> [(1,2),(3,5),(4,8),(6,7)]
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> [(1,2),(3,5),(4,7),(6,8)]
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> [(1,5),(2,3),(4,7),(6,8)]
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> [(1,5),(2,7),(3,4),(6,8)]
=> 2
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 0
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,13),(12,14)]
=> ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,12),(10,11),(13,14)]
=> [(1,2),(3,4),(5,6),(7,8),(9,11),(10,12),(13,14)]
=> ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> [(1,2),(3,4),(5,6),(7,8),(9,11),(10,13),(12,14)]
=> ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,13),(11,14)]
=> ? = 3
[(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10),(11,12),(13,14)]
=> ? = 1
[(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> [(1,2),(3,4),(5,6),(7,9),(8,10),(11,13),(12,14)]
=> ? = 2
[(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
=> [(1,2),(3,4),(5,6),(7,9),(8,11),(10,13),(12,14)]
=> ? = 3
[(1,2),(3,4),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> [(1,2),(3,4),(5,6),(7,10),(8,11),(9,12),(13,14)]
=> ? = 3
[(1,2),(3,4),(5,6),(7,14),(8,11),(9,10),(12,13)]
=> [(1,2),(3,4),(5,6),(7,10),(8,11),(9,13),(12,14)]
=> ? = 4
[(1,2),(3,4),(5,6),(7,14),(8,13),(9,10),(11,12)]
=> [(1,2),(3,4),(5,6),(7,10),(8,12),(9,13),(11,14)]
=> ? = 5
[(1,2),(3,4),(5,6),(7,14),(8,13),(9,12),(10,11)]
=> [(1,2),(3,4),(5,6),(7,11),(8,12),(9,13),(10,14)]
=> ? = 6
[(1,2),(3,4),(5,8),(6,7),(9,10),(11,12),(13,14)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10),(11,12),(13,14)]
=> ? = 1
[(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> [(1,2),(3,4),(5,7),(6,8),(9,10),(11,13),(12,14)]
=> ? = 2
[(1,2),(3,4),(5,8),(6,7),(9,12),(10,11),(13,14)]
=> [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12),(13,14)]
=> ? = 2
[(1,2),(3,4),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> [(1,2),(3,4),(5,7),(6,8),(9,12),(10,13),(11,14)]
=> ? = 4
[(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
=> [(1,2),(3,4),(5,7),(6,9),(8,11),(10,13),(12,14)]
=> ? = 4
[(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10),(11,12),(13,14)]
=> ? = 3
[(1,2),(3,4),(5,10),(6,9),(7,8),(11,14),(12,13)]
=> [(1,2),(3,4),(5,8),(6,9),(7,10),(11,13),(12,14)]
=> ? = 4
[(1,2),(3,4),(5,14),(6,9),(7,8),(10,11),(12,13)]
=> [(1,2),(3,4),(5,8),(6,9),(7,11),(10,13),(12,14)]
=> ? = 5
[(1,2),(3,4),(5,14),(6,11),(7,8),(9,10),(12,13)]
=> [(1,2),(3,4),(5,8),(6,10),(7,11),(9,13),(12,14)]
=> ? = 6
[(1,2),(3,4),(5,14),(6,13),(7,8),(9,10),(11,12)]
=> [(1,2),(3,4),(5,8),(6,10),(7,12),(9,13),(11,14)]
=> ? = 7
[(1,2),(3,4),(5,14),(6,13),(7,8),(9,12),(10,11)]
=> [(1,2),(3,4),(5,8),(6,11),(7,12),(9,13),(10,14)]
=> ? = 8
[(1,2),(3,4),(5,12),(6,11),(7,10),(8,9),(13,14)]
=> [(1,2),(3,4),(5,9),(6,10),(7,11),(8,12),(13,14)]
=> ? = 6
[(1,2),(3,4),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> [(1,2),(3,4),(5,9),(6,10),(7,11),(8,13),(12,14)]
=> ? = 7
[(1,2),(3,4),(5,14),(6,13),(7,12),(8,9),(10,11)]
=> [(1,2),(3,4),(5,9),(6,11),(7,12),(8,13),(10,14)]
=> ? = 9
[(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> [(1,2),(3,4),(5,10),(6,11),(7,12),(8,13),(9,14)]
=> ? = 10
[(1,2),(3,6),(4,5),(7,8),(9,10),(11,12),(13,14)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10),(11,12),(13,14)]
=> ? = 1
[(1,2),(3,6),(4,5),(7,8),(9,10),(11,14),(12,13)]
=> [(1,2),(3,5),(4,6),(7,8),(9,10),(11,13),(12,14)]
=> ? = 2
[(1,2),(3,6),(4,5),(7,8),(9,12),(10,11),(13,14)]
=> [(1,2),(3,5),(4,6),(7,8),(9,11),(10,12),(13,14)]
=> ? = 2
[(1,2),(3,6),(4,5),(7,8),(9,14),(10,13),(11,12)]
=> [(1,2),(3,5),(4,6),(7,8),(9,12),(10,13),(11,14)]
=> ? = 4
[(1,2),(3,6),(4,5),(7,10),(8,9),(11,12),(13,14)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10),(11,12),(13,14)]
=> ? = 2
[(1,2),(3,6),(4,5),(7,10),(8,9),(11,14),(12,13)]
=> [(1,2),(3,5),(4,6),(7,9),(8,10),(11,13),(12,14)]
=> ? = 3
[(1,2),(3,6),(4,5),(7,12),(8,11),(9,10),(13,14)]
=> [(1,2),(3,5),(4,6),(7,10),(8,11),(9,12),(13,14)]
=> ? = 4
[(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> [(1,2),(3,5),(4,6),(7,11),(8,12),(9,13),(10,14)]
=> ? = 7
[(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
=> [(1,2),(3,5),(4,7),(6,9),(8,11),(10,13),(12,14)]
=> ? = 5
[(1,2),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> [(1,2),(3,5),(4,10),(6,11),(7,12),(8,13),(9,14)]
=> ? = 11
[(1,2),(3,8),(4,7),(5,6),(9,10),(11,12),(13,14)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10),(11,12),(13,14)]
=> ? = 3
[(1,2),(3,8),(4,7),(5,6),(9,10),(11,14),(12,13)]
=> [(1,2),(3,6),(4,7),(5,8),(9,10),(11,13),(12,14)]
=> ? = 4
[(1,2),(3,8),(4,7),(5,6),(9,12),(10,11),(13,14)]
=> [(1,2),(3,6),(4,7),(5,8),(9,11),(10,12),(13,14)]
=> ? = 4
[(1,2),(3,8),(4,7),(5,6),(9,14),(10,13),(11,12)]
=> [(1,2),(3,6),(4,7),(5,8),(9,12),(10,13),(11,14)]
=> ? = 6
[(1,2),(3,14),(4,7),(5,6),(8,9),(10,11),(12,13)]
=> [(1,2),(3,6),(4,7),(5,9),(8,11),(10,13),(12,14)]
=> ? = 6
[(1,2),(3,14),(4,9),(5,6),(7,8),(10,11),(12,13)]
=> [(1,2),(3,6),(4,8),(5,9),(7,11),(10,13),(12,14)]
=> ? = 7
[(1,2),(3,14),(4,11),(5,6),(7,8),(9,10),(12,13)]
=> [(1,2),(3,6),(4,8),(5,10),(7,11),(9,13),(12,14)]
=> ? = 8
[(1,2),(3,14),(4,11),(5,6),(7,10),(8,9),(12,13)]
=> [(1,2),(3,6),(4,9),(5,10),(7,11),(8,13),(12,14)]
=> ? = 9
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,9),(10,11)]
=> [(1,2),(3,6),(4,9),(5,11),(7,12),(8,13),(10,14)]
=> ? = 11
[(1,2),(3,14),(4,13),(5,6),(7,12),(8,11),(9,10)]
=> [(1,2),(3,6),(4,10),(5,11),(7,12),(8,13),(9,14)]
=> ? = 12
[(1,2),(3,10),(4,9),(5,8),(6,7),(11,12),(13,14)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10),(11,12),(13,14)]
=> ? = 6
[(1,2),(3,10),(4,9),(5,8),(6,7),(11,14),(12,13)]
=> [(1,2),(3,7),(4,8),(5,9),(6,10),(11,13),(12,14)]
=> ? = 7
[(1,2),(3,14),(4,9),(5,8),(6,7),(10,11),(12,13)]
=> [(1,2),(3,7),(4,8),(5,9),(6,11),(10,13),(12,14)]
=> ? = 8
Description
The number of crossings of a perfect matching.
This is the number of pairs of edges ((a,b),(c,d)) such that a≤c≤b≤d, i.e., the edges (a,b) and (c,d) cross when drawing the perfect matching as a chord diagram.
The generating function for perfect matchings M of {1,…,2n} according to the number of crossings cr(M) is given by the Touchard-Riordan formula ([2], [4], a bijective proof is given in [7]):
\sum_{M} q^{\textrm{cr}(M)}
= \frac{1}{(1-q)^n} \sum_{k=0}^n\left(\binom{2n}{n-k}-\binom{2n}{n-k-1}\right)(-1)^k q^{\binom{k+1}{2}}
Matching statistic: St000233
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 57%
St000233: Set partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 57%
Values
[(1,2)]
=> {{1,2}}
=> 0
[(1,2),(3,4)]
=> {{1,2},{3,4}}
=> 0
[(1,3),(2,4)]
=> {{1,3},{2,4}}
=> 0
[(1,4),(2,3)]
=> {{1,4},{2,3}}
=> 1
[(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> 0
[(1,3),(2,4),(5,6)]
=> {{1,3},{2,4},{5,6}}
=> 0
[(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> 1
[(1,5),(2,3),(4,6)]
=> {{1,5},{2,3},{4,6}}
=> 1
[(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> 2
[(1,6),(2,4),(3,5)]
=> {{1,6},{2,4},{3,5}}
=> 2
[(1,5),(2,4),(3,6)]
=> {{1,5},{2,4},{3,6}}
=> 1
[(1,4),(2,5),(3,6)]
=> {{1,4},{2,5},{3,6}}
=> 0
[(1,3),(2,5),(4,6)]
=> {{1,3},{2,5},{4,6}}
=> 0
[(1,2),(3,5),(4,6)]
=> {{1,2},{3,5},{4,6}}
=> 0
[(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> 1
[(1,3),(2,6),(4,5)]
=> {{1,3},{2,6},{4,5}}
=> 1
[(1,4),(2,6),(3,5)]
=> {{1,4},{2,6},{3,5}}
=> 1
[(1,5),(2,6),(3,4)]
=> {{1,5},{2,6},{3,4}}
=> 2
[(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 0
[(1,3),(2,4),(5,6),(7,8)]
=> {{1,3},{2,4},{5,6},{7,8}}
=> 0
[(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> {{1,5},{2,3},{4,6},{7,8}}
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> {{1,7},{2,3},{4,5},{6,8}}
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> 3
[(1,8),(2,4),(3,5),(6,7)]
=> {{1,8},{2,4},{3,5},{6,7}}
=> 3
[(1,7),(2,4),(3,5),(6,8)]
=> {{1,7},{2,4},{3,5},{6,8}}
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> {{1,6},{2,4},{3,5},{7,8}}
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> {{1,5},{2,4},{3,6},{7,8}}
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> {{1,4},{2,5},{3,6},{7,8}}
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> {{1,3},{2,5},{4,6},{7,8}}
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> {{1,2},{3,5},{4,6},{7,8}}
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> {{1,3},{2,6},{4,5},{7,8}}
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> {{1,4},{2,6},{3,5},{7,8}}
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> {{1,5},{2,6},{3,4},{7,8}}
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> {{1,7},{2,5},{3,4},{6,8}}
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> 4
[(1,8),(2,6),(3,4),(5,7)]
=> {{1,8},{2,6},{3,4},{5,7}}
=> 4
[(1,7),(2,6),(3,4),(5,8)]
=> {{1,7},{2,6},{3,4},{5,8}}
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> {{1,6},{2,7},{3,4},{5,8}}
=> 2
[(1,5),(2,7),(3,4),(6,8)]
=> {{1,5},{2,7},{3,4},{6,8}}
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> {{1,4},{2,7},{3,5},{6,8}}
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> {{1,3},{2,7},{4,5},{6,8}}
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> {{1,2},{3,7},{4,5},{6,8}}
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> {{1,3},{2,8},{4,5},{6,7}}
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> {{1,4},{2,8},{3,5},{6,7}}
=> 2
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> {{1,3},{2,4},{5,6},{7,8},{9,10}}
=> ? = 0
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> {{1,4},{2,5},{3,6},{7,8},{9,10}}
=> ? = 0
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> {{1,3},{2,5},{4,6},{7,8},{9,10}}
=> ? = 0
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> {{1,2},{3,5},{4,6},{7,8},{9,10}}
=> ? = 0
[(1,5),(2,6),(3,7),(4,8),(9,10)]
=> {{1,5},{2,6},{3,7},{4,8},{9,10}}
=> ? = 0
[(1,4),(2,6),(3,7),(5,8),(9,10)]
=> {{1,4},{2,6},{3,7},{5,8},{9,10}}
=> ? = 0
[(1,3),(2,6),(4,7),(5,8),(9,10)]
=> {{1,3},{2,6},{4,7},{5,8},{9,10}}
=> ? = 0
[(1,2),(3,6),(4,7),(5,8),(9,10)]
=> {{1,2},{3,6},{4,7},{5,8},{9,10}}
=> ? = 0
[(1,2),(3,5),(4,7),(6,8),(9,10)]
=> {{1,2},{3,5},{4,7},{6,8},{9,10}}
=> ? = 0
[(1,3),(2,5),(4,7),(6,8),(9,10)]
=> {{1,3},{2,5},{4,7},{6,8},{9,10}}
=> ? = 0
[(1,4),(2,5),(3,7),(6,8),(9,10)]
=> {{1,4},{2,5},{3,7},{6,8},{9,10}}
=> ? = 0
[(1,3),(2,4),(5,7),(6,8),(9,10)]
=> {{1,3},{2,4},{5,7},{6,8},{9,10}}
=> ? = 0
[(1,2),(3,4),(5,7),(6,8),(9,10)]
=> {{1,2},{3,4},{5,7},{6,8},{9,10}}
=> ? = 0
[(1,2),(3,7),(4,8),(5,9),(6,10)]
=> {{1,2},{3,7},{4,8},{5,9},{6,10}}
=> ? = 0
[(1,3),(2,7),(4,8),(5,9),(6,10)]
=> {{1,3},{2,7},{4,8},{5,9},{6,10}}
=> ? = 0
[(1,4),(2,7),(3,8),(5,9),(6,10)]
=> {{1,4},{2,7},{3,8},{5,9},{6,10}}
=> ? = 0
[(1,5),(2,7),(3,8),(4,9),(6,10)]
=> {{1,5},{2,7},{3,8},{4,9},{6,10}}
=> ? = 0
[(1,6),(2,7),(3,8),(4,9),(5,10)]
=> {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> ? = 0
[(1,5),(2,6),(3,8),(4,9),(7,10)]
=> {{1,5},{2,6},{3,8},{4,9},{7,10}}
=> ? = 0
[(1,4),(2,6),(3,8),(5,9),(7,10)]
=> {{1,4},{2,6},{3,8},{5,9},{7,10}}
=> ? = 0
[(1,3),(2,6),(4,8),(5,9),(7,10)]
=> {{1,3},{2,6},{4,8},{5,9},{7,10}}
=> ? = 0
[(1,2),(3,6),(4,8),(5,9),(7,10)]
=> {{1,2},{3,6},{4,8},{5,9},{7,10}}
=> ? = 0
[(1,2),(3,5),(4,8),(6,9),(7,10)]
=> {{1,2},{3,5},{4,8},{6,9},{7,10}}
=> ? = 0
[(1,3),(2,5),(4,8),(6,9),(7,10)]
=> {{1,3},{2,5},{4,8},{6,9},{7,10}}
=> ? = 0
[(1,4),(2,5),(3,8),(6,9),(7,10)]
=> {{1,4},{2,5},{3,8},{6,9},{7,10}}
=> ? = 0
[(1,3),(2,4),(5,8),(6,9),(7,10)]
=> {{1,3},{2,4},{5,8},{6,9},{7,10}}
=> ? = 0
[(1,2),(3,4),(5,8),(6,9),(7,10)]
=> {{1,2},{3,4},{5,8},{6,9},{7,10}}
=> ? = 0
[(1,2),(3,4),(5,7),(6,9),(8,10)]
=> {{1,2},{3,4},{5,7},{6,9},{8,10}}
=> ? = 0
[(1,3),(2,4),(5,7),(6,9),(8,10)]
=> {{1,3},{2,4},{5,7},{6,9},{8,10}}
=> ? = 0
[(1,4),(2,5),(3,7),(6,9),(8,10)]
=> {{1,4},{2,5},{3,7},{6,9},{8,10}}
=> ? = 0
[(1,3),(2,5),(4,7),(6,9),(8,10)]
=> {{1,3},{2,5},{4,7},{6,9},{8,10}}
=> ? = 0
[(1,2),(3,5),(4,7),(6,9),(8,10)]
=> {{1,2},{3,5},{4,7},{6,9},{8,10}}
=> ? = 0
[(1,2),(3,6),(4,7),(5,9),(8,10)]
=> {{1,2},{3,6},{4,7},{5,9},{8,10}}
=> ? = 0
[(1,3),(2,6),(4,7),(5,9),(8,10)]
=> {{1,3},{2,6},{4,7},{5,9},{8,10}}
=> ? = 0
[(1,4),(2,6),(3,7),(5,9),(8,10)]
=> {{1,4},{2,6},{3,7},{5,9},{8,10}}
=> ? = 0
[(1,5),(2,6),(3,7),(4,9),(8,10)]
=> {{1,5},{2,6},{3,7},{4,9},{8,10}}
=> ? = 0
[(1,2),(3,5),(4,6),(7,9),(8,10)]
=> {{1,2},{3,5},{4,6},{7,9},{8,10}}
=> ? = 0
[(1,3),(2,5),(4,6),(7,9),(8,10)]
=> {{1,3},{2,5},{4,6},{7,9},{8,10}}
=> ? = 0
[(1,4),(2,5),(3,6),(7,9),(8,10)]
=> {{1,4},{2,5},{3,6},{7,9},{8,10}}
=> ? = 0
[(1,3),(2,4),(5,6),(7,9),(8,10)]
=> {{1,3},{2,4},{5,6},{7,9},{8,10}}
=> ? = 0
[(1,2),(3,4),(5,6),(7,9),(8,10)]
=> {{1,2},{3,4},{5,6},{7,9},{8,10}}
=> ? = 0
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12},{13,14}}
=> ? = 0
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,14},{12,13}}
=> ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,12),(10,11),(13,14)]
=> {{1,2},{3,4},{5,6},{7,8},{9,12},{10,11},{13,14}}
=> ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> {{1,2},{3,4},{5,6},{7,8},{9,14},{10,11},{12,13}}
=> ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> {{1,2},{3,4},{5,6},{7,8},{9,14},{10,13},{11,12}}
=> ? = 3
[(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9},{11,12},{13,14}}
=> ? = 1
[(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9},{11,14},{12,13}}
=> ? = 2
[(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
=> {{1,2},{3,4},{5,6},{7,14},{8,9},{10,11},{12,13}}
=> ? = 3
[(1,2),(3,4),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> {{1,2},{3,4},{5,6},{7,12},{8,11},{9,10},{13,14}}
=> ? = 3
Description
The number of nestings of a set partition.
This is given by the number of i < i' < j' < j such that i,j are two consecutive entries on one block, and i',j' are consecutive entries in another block.
Matching statistic: St000496
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 57%
St000496: Set partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 57%
Values
[(1,2)]
=> {{1,2}}
=> 0
[(1,2),(3,4)]
=> {{1,2},{3,4}}
=> 0
[(1,3),(2,4)]
=> {{1,3},{2,4}}
=> 0
[(1,4),(2,3)]
=> {{1,4},{2,3}}
=> 1
[(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> 0
[(1,3),(2,4),(5,6)]
=> {{1,3},{2,4},{5,6}}
=> 0
[(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> 1
[(1,5),(2,3),(4,6)]
=> {{1,5},{2,3},{4,6}}
=> 1
[(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> 2
[(1,6),(2,4),(3,5)]
=> {{1,6},{2,4},{3,5}}
=> 2
[(1,5),(2,4),(3,6)]
=> {{1,5},{2,4},{3,6}}
=> 1
[(1,4),(2,5),(3,6)]
=> {{1,4},{2,5},{3,6}}
=> 0
[(1,3),(2,5),(4,6)]
=> {{1,3},{2,5},{4,6}}
=> 0
[(1,2),(3,5),(4,6)]
=> {{1,2},{3,5},{4,6}}
=> 0
[(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> 1
[(1,3),(2,6),(4,5)]
=> {{1,3},{2,6},{4,5}}
=> 1
[(1,4),(2,6),(3,5)]
=> {{1,4},{2,6},{3,5}}
=> 1
[(1,5),(2,6),(3,4)]
=> {{1,5},{2,6},{3,4}}
=> 2
[(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 0
[(1,3),(2,4),(5,6),(7,8)]
=> {{1,3},{2,4},{5,6},{7,8}}
=> 0
[(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> {{1,5},{2,3},{4,6},{7,8}}
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> {{1,7},{2,3},{4,5},{6,8}}
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> 3
[(1,8),(2,4),(3,5),(6,7)]
=> {{1,8},{2,4},{3,5},{6,7}}
=> 3
[(1,7),(2,4),(3,5),(6,8)]
=> {{1,7},{2,4},{3,5},{6,8}}
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> {{1,6},{2,4},{3,5},{7,8}}
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> {{1,5},{2,4},{3,6},{7,8}}
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> {{1,4},{2,5},{3,6},{7,8}}
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> {{1,3},{2,5},{4,6},{7,8}}
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> {{1,2},{3,5},{4,6},{7,8}}
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> {{1,3},{2,6},{4,5},{7,8}}
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> {{1,4},{2,6},{3,5},{7,8}}
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> {{1,5},{2,6},{3,4},{7,8}}
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> {{1,7},{2,5},{3,4},{6,8}}
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> 4
[(1,8),(2,6),(3,4),(5,7)]
=> {{1,8},{2,6},{3,4},{5,7}}
=> 4
[(1,7),(2,6),(3,4),(5,8)]
=> {{1,7},{2,6},{3,4},{5,8}}
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> {{1,6},{2,7},{3,4},{5,8}}
=> 2
[(1,5),(2,7),(3,4),(6,8)]
=> {{1,5},{2,7},{3,4},{6,8}}
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> {{1,4},{2,7},{3,5},{6,8}}
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> {{1,3},{2,7},{4,5},{6,8}}
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> {{1,2},{3,7},{4,5},{6,8}}
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> {{1,3},{2,8},{4,5},{6,7}}
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> {{1,4},{2,8},{3,5},{6,7}}
=> 2
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> {{1,3},{2,4},{5,6},{7,8},{9,10}}
=> ? = 0
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> {{1,4},{2,5},{3,6},{7,8},{9,10}}
=> ? = 0
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> {{1,3},{2,5},{4,6},{7,8},{9,10}}
=> ? = 0
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> {{1,2},{3,5},{4,6},{7,8},{9,10}}
=> ? = 0
[(1,5),(2,6),(3,7),(4,8),(9,10)]
=> {{1,5},{2,6},{3,7},{4,8},{9,10}}
=> ? = 0
[(1,4),(2,6),(3,7),(5,8),(9,10)]
=> {{1,4},{2,6},{3,7},{5,8},{9,10}}
=> ? = 0
[(1,3),(2,6),(4,7),(5,8),(9,10)]
=> {{1,3},{2,6},{4,7},{5,8},{9,10}}
=> ? = 0
[(1,2),(3,6),(4,7),(5,8),(9,10)]
=> {{1,2},{3,6},{4,7},{5,8},{9,10}}
=> ? = 0
[(1,2),(3,5),(4,7),(6,8),(9,10)]
=> {{1,2},{3,5},{4,7},{6,8},{9,10}}
=> ? = 0
[(1,3),(2,5),(4,7),(6,8),(9,10)]
=> {{1,3},{2,5},{4,7},{6,8},{9,10}}
=> ? = 0
[(1,4),(2,5),(3,7),(6,8),(9,10)]
=> {{1,4},{2,5},{3,7},{6,8},{9,10}}
=> ? = 0
[(1,3),(2,4),(5,7),(6,8),(9,10)]
=> {{1,3},{2,4},{5,7},{6,8},{9,10}}
=> ? = 0
[(1,2),(3,4),(5,7),(6,8),(9,10)]
=> {{1,2},{3,4},{5,7},{6,8},{9,10}}
=> ? = 0
[(1,2),(3,7),(4,8),(5,9),(6,10)]
=> {{1,2},{3,7},{4,8},{5,9},{6,10}}
=> ? = 0
[(1,3),(2,7),(4,8),(5,9),(6,10)]
=> {{1,3},{2,7},{4,8},{5,9},{6,10}}
=> ? = 0
[(1,4),(2,7),(3,8),(5,9),(6,10)]
=> {{1,4},{2,7},{3,8},{5,9},{6,10}}
=> ? = 0
[(1,5),(2,7),(3,8),(4,9),(6,10)]
=> {{1,5},{2,7},{3,8},{4,9},{6,10}}
=> ? = 0
[(1,6),(2,7),(3,8),(4,9),(5,10)]
=> {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> ? = 0
[(1,5),(2,6),(3,8),(4,9),(7,10)]
=> {{1,5},{2,6},{3,8},{4,9},{7,10}}
=> ? = 0
[(1,4),(2,6),(3,8),(5,9),(7,10)]
=> {{1,4},{2,6},{3,8},{5,9},{7,10}}
=> ? = 0
[(1,3),(2,6),(4,8),(5,9),(7,10)]
=> {{1,3},{2,6},{4,8},{5,9},{7,10}}
=> ? = 0
[(1,2),(3,6),(4,8),(5,9),(7,10)]
=> {{1,2},{3,6},{4,8},{5,9},{7,10}}
=> ? = 0
[(1,2),(3,5),(4,8),(6,9),(7,10)]
=> {{1,2},{3,5},{4,8},{6,9},{7,10}}
=> ? = 0
[(1,3),(2,5),(4,8),(6,9),(7,10)]
=> {{1,3},{2,5},{4,8},{6,9},{7,10}}
=> ? = 0
[(1,4),(2,5),(3,8),(6,9),(7,10)]
=> {{1,4},{2,5},{3,8},{6,9},{7,10}}
=> ? = 0
[(1,3),(2,4),(5,8),(6,9),(7,10)]
=> {{1,3},{2,4},{5,8},{6,9},{7,10}}
=> ? = 0
[(1,2),(3,4),(5,8),(6,9),(7,10)]
=> {{1,2},{3,4},{5,8},{6,9},{7,10}}
=> ? = 0
[(1,2),(3,4),(5,7),(6,9),(8,10)]
=> {{1,2},{3,4},{5,7},{6,9},{8,10}}
=> ? = 0
[(1,3),(2,4),(5,7),(6,9),(8,10)]
=> {{1,3},{2,4},{5,7},{6,9},{8,10}}
=> ? = 0
[(1,4),(2,5),(3,7),(6,9),(8,10)]
=> {{1,4},{2,5},{3,7},{6,9},{8,10}}
=> ? = 0
[(1,3),(2,5),(4,7),(6,9),(8,10)]
=> {{1,3},{2,5},{4,7},{6,9},{8,10}}
=> ? = 0
[(1,2),(3,5),(4,7),(6,9),(8,10)]
=> {{1,2},{3,5},{4,7},{6,9},{8,10}}
=> ? = 0
[(1,2),(3,6),(4,7),(5,9),(8,10)]
=> {{1,2},{3,6},{4,7},{5,9},{8,10}}
=> ? = 0
[(1,3),(2,6),(4,7),(5,9),(8,10)]
=> {{1,3},{2,6},{4,7},{5,9},{8,10}}
=> ? = 0
[(1,4),(2,6),(3,7),(5,9),(8,10)]
=> {{1,4},{2,6},{3,7},{5,9},{8,10}}
=> ? = 0
[(1,5),(2,6),(3,7),(4,9),(8,10)]
=> {{1,5},{2,6},{3,7},{4,9},{8,10}}
=> ? = 0
[(1,2),(3,5),(4,6),(7,9),(8,10)]
=> {{1,2},{3,5},{4,6},{7,9},{8,10}}
=> ? = 0
[(1,3),(2,5),(4,6),(7,9),(8,10)]
=> {{1,3},{2,5},{4,6},{7,9},{8,10}}
=> ? = 0
[(1,4),(2,5),(3,6),(7,9),(8,10)]
=> {{1,4},{2,5},{3,6},{7,9},{8,10}}
=> ? = 0
[(1,3),(2,4),(5,6),(7,9),(8,10)]
=> {{1,3},{2,4},{5,6},{7,9},{8,10}}
=> ? = 0
[(1,2),(3,4),(5,6),(7,9),(8,10)]
=> {{1,2},{3,4},{5,6},{7,9},{8,10}}
=> ? = 0
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,12},{13,14}}
=> ? = 0
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
=> {{1,2},{3,4},{5,6},{7,8},{9,10},{11,14},{12,13}}
=> ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,12),(10,11),(13,14)]
=> {{1,2},{3,4},{5,6},{7,8},{9,12},{10,11},{13,14}}
=> ? = 1
[(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
=> {{1,2},{3,4},{5,6},{7,8},{9,14},{10,11},{12,13}}
=> ? = 2
[(1,2),(3,4),(5,6),(7,8),(9,14),(10,13),(11,12)]
=> {{1,2},{3,4},{5,6},{7,8},{9,14},{10,13},{11,12}}
=> ? = 3
[(1,2),(3,4),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9},{11,12},{13,14}}
=> ? = 1
[(1,2),(3,4),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> {{1,2},{3,4},{5,6},{7,10},{8,9},{11,14},{12,13}}
=> ? = 2
[(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
=> {{1,2},{3,4},{5,6},{7,14},{8,9},{10,11},{12,13}}
=> ? = 3
[(1,2),(3,4),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> {{1,2},{3,4},{5,6},{7,12},{8,11},{9,10},{13,14}}
=> ? = 3
Description
The rcs statistic of a set partition.
Let S = B_1,\ldots,B_k be a set partition with ordered blocks B_i and with \operatorname{min} B_a < \operatorname{min} B_b for a < b.
According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of S is given by a pair i > j such that j = \operatorname{max} B_b and i \in B_a for a < b.
Matching statistic: St000232
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Values
[(1,2)]
=> {{1,2}}
=> {{1,2}}
=> 0
[(1,2),(3,4)]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
[(1,3),(2,4)]
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 0
[(1,4),(2,3)]
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 1
[(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> {{1,2},{3,4},{5,6}}
=> 0
[(1,3),(2,4),(5,6)]
=> {{1,3},{2,4},{5,6}}
=> {{1,4},{2,3},{5,6}}
=> 0
[(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> {{1,3},{2,4},{5,6}}
=> 1
[(1,5),(2,3),(4,6)]
=> {{1,5},{2,3},{4,6}}
=> {{1,3},{2,6},{4,5}}
=> 1
[(1,6),(2,3),(4,5)]
=> {{1,6},{2,3},{4,5}}
=> {{1,3},{2,5},{4,6}}
=> 2
[(1,6),(2,4),(3,5)]
=> {{1,6},{2,4},{3,5}}
=> {{1,5},{2,4},{3,6}}
=> 2
[(1,5),(2,4),(3,6)]
=> {{1,5},{2,4},{3,6}}
=> {{1,6},{2,4},{3,5}}
=> 1
[(1,4),(2,5),(3,6)]
=> {{1,4},{2,5},{3,6}}
=> {{1,6},{2,5},{3,4}}
=> 0
[(1,3),(2,5),(4,6)]
=> {{1,3},{2,5},{4,6}}
=> {{1,6},{2,3},{4,5}}
=> 0
[(1,2),(3,5),(4,6)]
=> {{1,2},{3,5},{4,6}}
=> {{1,2},{3,6},{4,5}}
=> 0
[(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> {{1,2},{3,5},{4,6}}
=> 1
[(1,3),(2,6),(4,5)]
=> {{1,3},{2,6},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> 1
[(1,4),(2,6),(3,5)]
=> {{1,4},{2,6},{3,5}}
=> {{1,5},{2,6},{3,4}}
=> 1
[(1,5),(2,6),(3,4)]
=> {{1,5},{2,6},{3,4}}
=> {{1,4},{2,6},{3,5}}
=> 2
[(1,6),(2,5),(3,4)]
=> {{1,6},{2,5},{3,4}}
=> {{1,4},{2,5},{3,6}}
=> 3
[(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2},{3,4},{5,6},{7,8}}
=> 0
[(1,3),(2,4),(5,6),(7,8)]
=> {{1,3},{2,4},{5,6},{7,8}}
=> {{1,4},{2,3},{5,6},{7,8}}
=> 0
[(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> {{1,3},{2,4},{5,6},{7,8}}
=> 1
[(1,5),(2,3),(4,6),(7,8)]
=> {{1,5},{2,3},{4,6},{7,8}}
=> {{1,3},{2,6},{4,5},{7,8}}
=> 1
[(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> {{1,3},{2,5},{4,6},{7,8}}
=> 2
[(1,7),(2,3),(4,5),(6,8)]
=> {{1,7},{2,3},{4,5},{6,8}}
=> {{1,3},{2,5},{4,8},{6,7}}
=> 2
[(1,8),(2,3),(4,5),(6,7)]
=> {{1,8},{2,3},{4,5},{6,7}}
=> {{1,3},{2,5},{4,7},{6,8}}
=> 3
[(1,8),(2,4),(3,5),(6,7)]
=> {{1,8},{2,4},{3,5},{6,7}}
=> {{1,5},{2,4},{3,7},{6,8}}
=> 3
[(1,7),(2,4),(3,5),(6,8)]
=> {{1,7},{2,4},{3,5},{6,8}}
=> {{1,5},{2,4},{3,8},{6,7}}
=> 2
[(1,6),(2,4),(3,5),(7,8)]
=> {{1,6},{2,4},{3,5},{7,8}}
=> {{1,5},{2,4},{3,6},{7,8}}
=> 2
[(1,5),(2,4),(3,6),(7,8)]
=> {{1,5},{2,4},{3,6},{7,8}}
=> {{1,6},{2,4},{3,5},{7,8}}
=> 1
[(1,4),(2,5),(3,6),(7,8)]
=> {{1,4},{2,5},{3,6},{7,8}}
=> {{1,6},{2,5},{3,4},{7,8}}
=> 0
[(1,3),(2,5),(4,6),(7,8)]
=> {{1,3},{2,5},{4,6},{7,8}}
=> {{1,6},{2,3},{4,5},{7,8}}
=> 0
[(1,2),(3,5),(4,6),(7,8)]
=> {{1,2},{3,5},{4,6},{7,8}}
=> {{1,2},{3,6},{4,5},{7,8}}
=> 0
[(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> {{1,2},{3,5},{4,6},{7,8}}
=> 1
[(1,3),(2,6),(4,5),(7,8)]
=> {{1,3},{2,6},{4,5},{7,8}}
=> {{1,5},{2,3},{4,6},{7,8}}
=> 1
[(1,4),(2,6),(3,5),(7,8)]
=> {{1,4},{2,6},{3,5},{7,8}}
=> {{1,5},{2,6},{3,4},{7,8}}
=> 1
[(1,5),(2,6),(3,4),(7,8)]
=> {{1,5},{2,6},{3,4},{7,8}}
=> {{1,4},{2,6},{3,5},{7,8}}
=> 2
[(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> {{1,4},{2,5},{3,6},{7,8}}
=> 3
[(1,7),(2,5),(3,4),(6,8)]
=> {{1,7},{2,5},{3,4},{6,8}}
=> {{1,4},{2,5},{3,8},{6,7}}
=> 3
[(1,8),(2,5),(3,4),(6,7)]
=> {{1,8},{2,5},{3,4},{6,7}}
=> {{1,4},{2,5},{3,7},{6,8}}
=> 4
[(1,8),(2,6),(3,4),(5,7)]
=> {{1,8},{2,6},{3,4},{5,7}}
=> {{1,4},{2,7},{3,6},{5,8}}
=> 4
[(1,7),(2,6),(3,4),(5,8)]
=> {{1,7},{2,6},{3,4},{5,8}}
=> {{1,4},{2,8},{3,6},{5,7}}
=> 3
[(1,6),(2,7),(3,4),(5,8)]
=> {{1,6},{2,7},{3,4},{5,8}}
=> {{1,4},{2,8},{3,7},{5,6}}
=> 2
[(1,5),(2,7),(3,4),(6,8)]
=> {{1,5},{2,7},{3,4},{6,8}}
=> {{1,4},{2,8},{3,5},{6,7}}
=> 2
[(1,4),(2,7),(3,5),(6,8)]
=> {{1,4},{2,7},{3,5},{6,8}}
=> {{1,5},{2,8},{3,4},{6,7}}
=> 1
[(1,3),(2,7),(4,5),(6,8)]
=> {{1,3},{2,7},{4,5},{6,8}}
=> {{1,5},{2,3},{4,8},{6,7}}
=> 1
[(1,2),(3,7),(4,5),(6,8)]
=> {{1,2},{3,7},{4,5},{6,8}}
=> {{1,2},{3,5},{4,8},{6,7}}
=> 1
[(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> {{1,2},{3,5},{4,7},{6,8}}
=> 2
[(1,3),(2,8),(4,5),(6,7)]
=> {{1,3},{2,8},{4,5},{6,7}}
=> {{1,5},{2,3},{4,7},{6,8}}
=> 2
[(1,4),(2,8),(3,5),(6,7)]
=> {{1,4},{2,8},{3,5},{6,7}}
=> {{1,5},{2,7},{3,4},{6,8}}
=> 2
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> {{1,3},{2,4},{5,6},{7,8},{9,10}}
=> ?
=> ? = 0
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> {{1,4},{2,3},{5,6},{7,8},{9,10}}
=> {{1,3},{2,4},{5,6},{7,8},{9,10}}
=> ? = 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> {{1,6},{2,3},{4,5},{7,8},{9,10}}
=> {{1,3},{2,5},{4,6},{7,8},{9,10}}
=> ? = 2
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> {{1,8},{2,3},{4,5},{6,7},{9,10}}
=> {{1,3},{2,5},{4,7},{6,8},{9,10}}
=> ? = 3
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> {{1,10},{2,3},{4,5},{6,7},{8,9}}
=> {{1,3},{2,5},{4,7},{6,9},{8,10}}
=> ? = 4
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> {{1,4},{2,5},{3,6},{7,8},{9,10}}
=> ?
=> ? = 0
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> {{1,3},{2,5},{4,6},{7,8},{9,10}}
=> ?
=> ? = 0
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> {{1,2},{3,5},{4,6},{7,8},{9,10}}
=> ?
=> ? = 0
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> {{1,2},{3,6},{4,5},{7,8},{9,10}}
=> {{1,2},{3,5},{4,6},{7,8},{9,10}}
=> ? = 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> {{1,6},{2,5},{3,4},{7,8},{9,10}}
=> {{1,4},{2,5},{3,6},{7,8},{9,10}}
=> ? = 3
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> {{1,8},{2,5},{3,4},{6,7},{9,10}}
=> {{1,4},{2,5},{3,7},{6,8},{9,10}}
=> ? = 4
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> {{1,10},{2,5},{3,4},{6,7},{8,9}}
=> {{1,4},{2,5},{3,7},{6,9},{8,10}}
=> ? = 5
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> {{1,2},{3,8},{4,5},{6,7},{9,10}}
=> {{1,2},{3,5},{4,7},{6,8},{9,10}}
=> ? = 2
[(1,8),(2,7),(3,4),(5,6),(9,10)]
=> {{1,8},{2,7},{3,4},{5,6},{9,10}}
=> {{1,4},{2,6},{3,7},{5,8},{9,10}}
=> ? = 5
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> {{1,10},{2,7},{3,4},{5,6},{8,9}}
=> {{1,4},{2,6},{3,7},{5,9},{8,10}}
=> ? = 6
[(1,2),(3,10),(4,5),(6,7),(8,9)]
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> {{1,2},{3,5},{4,7},{6,9},{8,10}}
=> ? = 3
[(1,10),(2,9),(3,4),(5,6),(7,8)]
=> {{1,10},{2,9},{3,4},{5,6},{7,8}}
=> {{1,4},{2,6},{3,8},{5,9},{7,10}}
=> ? = 7
[(1,5),(2,6),(3,7),(4,8),(9,10)]
=> {{1,5},{2,6},{3,7},{4,8},{9,10}}
=> ?
=> ? = 0
[(1,4),(2,6),(3,7),(5,8),(9,10)]
=> {{1,4},{2,6},{3,7},{5,8},{9,10}}
=> ?
=> ? = 0
[(1,3),(2,6),(4,7),(5,8),(9,10)]
=> {{1,3},{2,6},{4,7},{5,8},{9,10}}
=> ?
=> ? = 0
[(1,2),(3,6),(4,7),(5,8),(9,10)]
=> {{1,2},{3,6},{4,7},{5,8},{9,10}}
=> ?
=> ? = 0
[(1,2),(3,5),(4,7),(6,8),(9,10)]
=> {{1,2},{3,5},{4,7},{6,8},{9,10}}
=> ?
=> ? = 0
[(1,3),(2,5),(4,7),(6,8),(9,10)]
=> {{1,3},{2,5},{4,7},{6,8},{9,10}}
=> ?
=> ? = 0
[(1,4),(2,5),(3,7),(6,8),(9,10)]
=> {{1,4},{2,5},{3,7},{6,8},{9,10}}
=> ?
=> ? = 0
[(1,3),(2,4),(5,7),(6,8),(9,10)]
=> {{1,3},{2,4},{5,7},{6,8},{9,10}}
=> ?
=> ? = 0
[(1,2),(3,4),(5,7),(6,8),(9,10)]
=> {{1,2},{3,4},{5,7},{6,8},{9,10}}
=> ?
=> ? = 0
[(1,2),(3,4),(5,8),(6,7),(9,10)]
=> {{1,2},{3,4},{5,8},{6,7},{9,10}}
=> {{1,2},{3,4},{5,7},{6,8},{9,10}}
=> ? = 1
[(1,4),(2,3),(5,8),(6,7),(9,10)]
=> {{1,4},{2,3},{5,8},{6,7},{9,10}}
=> {{1,3},{2,4},{5,7},{6,8},{9,10}}
=> ? = 2
[(1,8),(2,3),(4,7),(5,6),(9,10)]
=> {{1,8},{2,3},{4,7},{5,6},{9,10}}
=> {{1,3},{2,6},{4,7},{5,8},{9,10}}
=> ? = 4
[(1,10),(2,3),(4,7),(5,6),(8,9)]
=> {{1,10},{2,3},{4,7},{5,6},{8,9}}
=> {{1,3},{2,6},{4,7},{5,9},{8,10}}
=> ? = 5
[(1,2),(3,8),(4,7),(5,6),(9,10)]
=> {{1,2},{3,8},{4,7},{5,6},{9,10}}
=> {{1,2},{3,6},{4,7},{5,8},{9,10}}
=> ? = 3
[(1,8),(2,7),(3,6),(4,5),(9,10)]
=> {{1,8},{2,7},{3,6},{4,5},{9,10}}
=> {{1,5},{2,6},{3,7},{4,8},{9,10}}
=> ? = 6
[(1,10),(2,7),(3,6),(4,5),(8,9)]
=> {{1,10},{2,7},{3,6},{4,5},{8,9}}
=> {{1,5},{2,6},{3,7},{4,9},{8,10}}
=> ? = 7
[(1,2),(3,10),(4,7),(5,6),(8,9)]
=> {{1,2},{3,10},{4,7},{5,6},{8,9}}
=> {{1,2},{3,6},{4,7},{5,9},{8,10}}
=> ? = 4
[(1,10),(2,9),(3,6),(4,5),(7,8)]
=> {{1,10},{2,9},{3,6},{4,5},{7,8}}
=> {{1,5},{2,6},{3,8},{4,9},{7,10}}
=> ? = 8
[(1,2),(3,4),(5,10),(6,7),(8,9)]
=> {{1,2},{3,4},{5,10},{6,7},{8,9}}
=> {{1,2},{3,4},{5,7},{6,9},{8,10}}
=> ? = 2
[(1,4),(2,3),(5,10),(6,7),(8,9)]
=> {{1,4},{2,3},{5,10},{6,7},{8,9}}
=> {{1,3},{2,4},{5,7},{6,9},{8,10}}
=> ? = 3
[(1,10),(2,3),(4,9),(5,6),(7,8)]
=> {{1,10},{2,3},{4,9},{5,6},{7,8}}
=> {{1,3},{2,6},{4,8},{5,9},{7,10}}
=> ? = 6
[(1,2),(3,10),(4,9),(5,6),(7,8)]
=> {{1,2},{3,10},{4,9},{5,6},{7,8}}
=> {{1,2},{3,6},{4,8},{5,9},{7,10}}
=> ? = 5
[(1,10),(2,9),(3,8),(4,5),(6,7)]
=> {{1,10},{2,9},{3,8},{4,5},{6,7}}
=> {{1,5},{2,7},{3,8},{4,9},{6,10}}
=> ? = 9
[(1,2),(3,7),(4,8),(5,9),(6,10)]
=> {{1,2},{3,7},{4,8},{5,9},{6,10}}
=> ?
=> ? = 0
[(1,3),(2,7),(4,8),(5,9),(6,10)]
=> {{1,3},{2,7},{4,8},{5,9},{6,10}}
=> ?
=> ? = 0
[(1,4),(2,7),(3,8),(5,9),(6,10)]
=> {{1,4},{2,7},{3,8},{5,9},{6,10}}
=> ?
=> ? = 0
[(1,5),(2,7),(3,8),(4,9),(6,10)]
=> {{1,5},{2,7},{3,8},{4,9},{6,10}}
=> ?
=> ? = 0
[(1,5),(2,6),(3,8),(4,9),(7,10)]
=> {{1,5},{2,6},{3,8},{4,9},{7,10}}
=> ?
=> ? = 0
[(1,4),(2,6),(3,8),(5,9),(7,10)]
=> {{1,4},{2,6},{3,8},{5,9},{7,10}}
=> ?
=> ? = 0
[(1,3),(2,6),(4,8),(5,9),(7,10)]
=> {{1,3},{2,6},{4,8},{5,9},{7,10}}
=> ?
=> ? = 0
[(1,2),(3,6),(4,8),(5,9),(7,10)]
=> {{1,2},{3,6},{4,8},{5,9},{7,10}}
=> ?
=> ? = 0
[(1,2),(3,5),(4,8),(6,9),(7,10)]
=> {{1,2},{3,5},{4,8},{6,9},{7,10}}
=> ?
=> ? = 0
[(1,3),(2,5),(4,8),(6,9),(7,10)]
=> {{1,3},{2,5},{4,8},{6,9},{7,10}}
=> ?
=> ? = 0
Description
The number of crossings of a set partition.
This is given by the number of i < i' < j < j' such that i,j are two consecutive entries on one block, and i',j' are consecutive entries in another block.
Matching statistic: St000123
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Values
[(1,2)]
=> [2,1] => [2,1] => 0
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => 0
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,4,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => 0
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,4,1,6,5] => 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,5,3,6,1] => 2
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,2,5,1,6,3] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => 0
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,5,3,6,2] => 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,3,5,1,6,2] => 2
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => 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] => 0
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => 0
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => 2
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => 3
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,5,3,6,1,8,7] => 2
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => 0
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => 0
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => 0
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,1,5,3,6,2,8,7] => 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => 4
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => 4
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => 3
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => 2
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,1,5,3,7,2,8,6] => 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => 2
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => [3,1,4,2,6,5,8,7,10,9] => ? = 0
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [3,2,4,1,6,5,8,7,10,9] => ? = 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [3,2,5,4,6,1,8,7,10,9] => ? = 2
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [3,2,5,4,7,6,8,1,10,9] => ? = 3
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [3,2,5,4,7,6,9,8,10,1] => ? = 4
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => [4,1,5,2,6,3,8,7,10,9] => ? = 0
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => [3,1,5,2,6,4,8,7,10,9] => ? = 0
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => [2,1,5,3,6,4,8,7,10,9] => ? = 0
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,5,4,6,3,8,7,10,9] => ? = 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [4,3,5,2,6,1,8,7,10,9] => ? = 3
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [4,3,5,2,7,6,8,1,10,9] => ? = 4
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [4,3,5,2,7,6,9,8,10,1] => ? = 5
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,1,5,4,7,6,8,3,10,9] => ? = 2
[(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [4,3,6,5,7,2,8,1,10,9] => ? = 5
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [4,3,6,5,7,2,9,8,10,1] => ? = 6
[(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [2,1,5,4,7,6,9,8,10,3] => ? = 3
[(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [4,3,6,5,8,7,9,2,10,1] => ? = 7
[(1,5),(2,6),(3,7),(4,8),(9,10)]
=> [5,6,7,8,1,2,3,4,10,9] => [5,1,6,2,7,3,8,4,10,9] => ? = 0
[(1,4),(2,6),(3,7),(5,8),(9,10)]
=> [4,6,7,1,8,2,3,5,10,9] => [4,1,6,2,7,3,8,5,10,9] => ? = 0
[(1,3),(2,6),(4,7),(5,8),(9,10)]
=> [3,6,1,7,8,2,4,5,10,9] => [3,1,6,2,7,4,8,5,10,9] => ? = 0
[(1,2),(3,6),(4,7),(5,8),(9,10)]
=> [2,1,6,7,8,3,4,5,10,9] => [2,1,6,3,7,4,8,5,10,9] => ? = 0
[(1,2),(3,5),(4,7),(6,8),(9,10)]
=> [2,1,5,7,3,8,4,6,10,9] => [2,1,5,3,7,4,8,6,10,9] => ? = 0
[(1,3),(2,5),(4,7),(6,8),(9,10)]
=> [3,5,1,7,2,8,4,6,10,9] => [3,1,5,2,7,4,8,6,10,9] => ? = 0
[(1,4),(2,5),(3,7),(6,8),(9,10)]
=> [4,5,7,1,2,8,3,6,10,9] => [4,1,5,2,7,3,8,6,10,9] => ? = 0
[(1,3),(2,4),(5,7),(6,8),(9,10)]
=> [3,4,1,2,7,8,5,6,10,9] => [3,1,4,2,7,5,8,6,10,9] => ? = 0
[(1,2),(3,4),(5,7),(6,8),(9,10)]
=> [2,1,4,3,7,8,5,6,10,9] => [2,1,4,3,7,5,8,6,10,9] => ? = 0
[(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,7,6,8,5,10,9] => ? = 1
[(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [3,2,4,1,7,6,8,5,10,9] => ? = 2
[(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [3,2,6,5,7,4,8,1,10,9] => ? = 4
[(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [3,2,6,5,7,4,9,8,10,1] => ? = 5
[(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,6,5,7,4,8,3,10,9] => ? = 3
[(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [5,4,6,3,7,2,8,1,10,9] => ? = 6
[(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [5,4,6,3,7,2,9,8,10,1] => ? = 7
[(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [2,1,6,5,7,4,9,8,10,3] => ? = 4
[(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [5,4,6,3,8,7,9,2,10,1] => ? = 8
[(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,7,6,9,8,10,5] => ? = 2
[(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [3,2,4,1,7,6,9,8,10,5] => ? = 3
[(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [3,2,6,5,8,7,9,4,10,1] => ? = 6
[(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [2,1,6,5,8,7,9,4,10,3] => ? = 5
[(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [5,4,7,6,8,3,9,2,10,1] => ? = 9
[(1,2),(3,7),(4,8),(5,9),(6,10)]
=> [2,1,7,8,9,10,3,4,5,6] => [2,1,7,3,8,4,9,5,10,6] => ? = 0
[(1,3),(2,7),(4,8),(5,9),(6,10)]
=> [3,7,1,8,9,10,2,4,5,6] => [3,1,7,2,8,4,9,5,10,6] => ? = 0
[(1,4),(2,7),(3,8),(5,9),(6,10)]
=> [4,7,8,1,9,10,2,3,5,6] => [4,1,7,2,8,3,9,5,10,6] => ? = 0
[(1,5),(2,7),(3,8),(4,9),(6,10)]
=> [5,7,8,9,1,10,2,3,4,6] => [5,1,7,2,8,3,9,4,10,6] => ? = 0
[(1,6),(2,7),(3,8),(4,9),(5,10)]
=> [6,7,8,9,10,1,2,3,4,5] => [6,1,7,2,8,3,9,4,10,5] => ? = 0
[(1,5),(2,6),(3,8),(4,9),(7,10)]
=> [5,6,8,9,1,2,10,3,4,7] => [5,1,6,2,8,3,9,4,10,7] => ? = 0
[(1,4),(2,6),(3,8),(5,9),(7,10)]
=> [4,6,8,1,9,2,10,3,5,7] => [4,1,6,2,8,3,9,5,10,7] => ? = 0
[(1,3),(2,6),(4,8),(5,9),(7,10)]
=> [3,6,1,8,9,2,10,4,5,7] => [3,1,6,2,8,4,9,5,10,7] => ? = 0
[(1,2),(3,6),(4,8),(5,9),(7,10)]
=> [2,1,6,8,9,3,10,4,5,7] => [2,1,6,3,8,4,9,5,10,7] => ? = 0
[(1,2),(3,5),(4,8),(6,9),(7,10)]
=> [2,1,5,8,3,9,10,4,6,7] => [2,1,5,3,8,4,9,6,10,7] => ? = 0
Description
The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
The Simion-Schmidt map takes a permutation and turns each occcurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Matching statistic: St000359
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Values
[(1,2)]
=> [2,1] => [2,1] => 0
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)]
=> [3,4,1,2] => [3,1,4,2] => 0
[(1,4),(2,3)]
=> [4,3,2,1] => [3,2,4,1] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => 0
[(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [3,2,4,1,6,5] => 1
[(1,5),(2,3),(4,6)]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1
[(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [3,2,5,4,6,1] => 2
[(1,6),(2,4),(3,5)]
=> [6,4,5,2,3,1] => [4,2,5,3,6,1] => 2
[(1,5),(2,4),(3,6)]
=> [5,4,6,2,1,3] => [4,2,5,1,6,3] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,1,5,2,6,4] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => 0
[(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1
[(1,3),(2,6),(4,5)]
=> [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1
[(1,4),(2,6),(3,5)]
=> [4,6,5,1,3,2] => [4,1,5,3,6,2] => 1
[(1,5),(2,6),(3,4)]
=> [5,6,4,3,1,2] => [4,3,5,1,6,2] => 2
[(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => 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] => 0
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,1,4,2,6,5,8,7] => 0
[(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => 1
[(1,5),(2,3),(4,6),(7,8)]
=> [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => 1
[(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [3,2,5,4,6,1,8,7] => 2
[(1,7),(2,3),(4,5),(6,8)]
=> [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => 2
[(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => 3
[(1,7),(2,4),(3,5),(6,8)]
=> [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => 2
[(1,6),(2,4),(3,5),(7,8)]
=> [6,4,5,2,3,1,8,7] => [4,2,5,3,6,1,8,7] => 2
[(1,5),(2,4),(3,6),(7,8)]
=> [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => 0
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => 0
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => 0
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,6,5,1,3,2,8,7] => [4,1,5,3,6,2,8,7] => 1
[(1,5),(2,6),(3,4),(7,8)]
=> [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => 2
[(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => 3
[(1,7),(2,5),(3,4),(6,8)]
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => 3
[(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => 4
[(1,8),(2,6),(3,4),(5,7)]
=> [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => 4
[(1,7),(2,6),(3,4),(5,8)]
=> [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => 3
[(1,6),(2,7),(3,4),(5,8)]
=> [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => 2
[(1,5),(2,7),(3,4),(6,8)]
=> [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,7,5,1,3,8,2,6] => [4,1,5,3,7,2,8,6] => 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [2,1,5,4,7,6,8,3] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => 2
[(1,3),(2,4),(5,6),(7,8),(9,10)]
=> [3,4,1,2,6,5,8,7,10,9] => [3,1,4,2,6,5,8,7,10,9] => ? = 0
[(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [3,2,4,1,6,5,8,7,10,9] => ? = 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [3,2,5,4,6,1,8,7,10,9] => ? = 2
[(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [3,2,5,4,7,6,8,1,10,9] => ? = 3
[(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [3,2,5,4,7,6,9,8,10,1] => ? = 4
[(1,4),(2,5),(3,6),(7,8),(9,10)]
=> [4,5,6,1,2,3,8,7,10,9] => [4,1,5,2,6,3,8,7,10,9] => ? = 0
[(1,3),(2,5),(4,6),(7,8),(9,10)]
=> [3,5,1,6,2,4,8,7,10,9] => [3,1,5,2,6,4,8,7,10,9] => ? = 0
[(1,2),(3,5),(4,6),(7,8),(9,10)]
=> [2,1,5,6,3,4,8,7,10,9] => [2,1,5,3,6,4,8,7,10,9] => ? = 0
[(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [2,1,5,4,6,3,8,7,10,9] => ? = 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [4,3,5,2,6,1,8,7,10,9] => ? = 3
[(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [4,3,5,2,7,6,8,1,10,9] => ? = 4
[(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [4,3,5,2,7,6,9,8,10,1] => ? = 5
[(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [2,1,5,4,7,6,8,3,10,9] => ? = 2
[(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [4,3,6,5,7,2,8,1,10,9] => ? = 5
[(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [4,3,6,5,7,2,9,8,10,1] => ? = 6
[(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [2,1,5,4,7,6,9,8,10,3] => ? = 3
[(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [4,3,6,5,8,7,9,2,10,1] => ? = 7
[(1,5),(2,6),(3,7),(4,8),(9,10)]
=> [5,6,7,8,1,2,3,4,10,9] => [5,1,6,2,7,3,8,4,10,9] => ? = 0
[(1,4),(2,6),(3,7),(5,8),(9,10)]
=> [4,6,7,1,8,2,3,5,10,9] => [4,1,6,2,7,3,8,5,10,9] => ? = 0
[(1,3),(2,6),(4,7),(5,8),(9,10)]
=> [3,6,1,7,8,2,4,5,10,9] => [3,1,6,2,7,4,8,5,10,9] => ? = 0
[(1,2),(3,6),(4,7),(5,8),(9,10)]
=> [2,1,6,7,8,3,4,5,10,9] => [2,1,6,3,7,4,8,5,10,9] => ? = 0
[(1,2),(3,5),(4,7),(6,8),(9,10)]
=> [2,1,5,7,3,8,4,6,10,9] => [2,1,5,3,7,4,8,6,10,9] => ? = 0
[(1,3),(2,5),(4,7),(6,8),(9,10)]
=> [3,5,1,7,2,8,4,6,10,9] => [3,1,5,2,7,4,8,6,10,9] => ? = 0
[(1,4),(2,5),(3,7),(6,8),(9,10)]
=> [4,5,7,1,2,8,3,6,10,9] => [4,1,5,2,7,3,8,6,10,9] => ? = 0
[(1,3),(2,4),(5,7),(6,8),(9,10)]
=> [3,4,1,2,7,8,5,6,10,9] => [3,1,4,2,7,5,8,6,10,9] => ? = 0
[(1,2),(3,4),(5,7),(6,8),(9,10)]
=> [2,1,4,3,7,8,5,6,10,9] => [2,1,4,3,7,5,8,6,10,9] => ? = 0
[(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [2,1,4,3,7,6,8,5,10,9] => ? = 1
[(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [3,2,4,1,7,6,8,5,10,9] => ? = 2
[(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [3,2,6,5,7,4,8,1,10,9] => ? = 4
[(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [3,2,6,5,7,4,9,8,10,1] => ? = 5
[(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [2,1,6,5,7,4,8,3,10,9] => ? = 3
[(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [5,4,6,3,7,2,8,1,10,9] => ? = 6
[(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [5,4,6,3,7,2,9,8,10,1] => ? = 7
[(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [2,1,6,5,7,4,9,8,10,3] => ? = 4
[(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [5,4,6,3,8,7,9,2,10,1] => ? = 8
[(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [2,1,4,3,7,6,9,8,10,5] => ? = 2
[(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [3,2,4,1,7,6,9,8,10,5] => ? = 3
[(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [3,2,6,5,8,7,9,4,10,1] => ? = 6
[(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [2,1,6,5,8,7,9,4,10,3] => ? = 5
[(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [5,4,7,6,8,3,9,2,10,1] => ? = 9
[(1,2),(3,7),(4,8),(5,9),(6,10)]
=> [2,1,7,8,9,10,3,4,5,6] => [2,1,7,3,8,4,9,5,10,6] => ? = 0
[(1,3),(2,7),(4,8),(5,9),(6,10)]
=> [3,7,1,8,9,10,2,4,5,6] => [3,1,7,2,8,4,9,5,10,6] => ? = 0
[(1,4),(2,7),(3,8),(5,9),(6,10)]
=> [4,7,8,1,9,10,2,3,5,6] => [4,1,7,2,8,3,9,5,10,6] => ? = 0
[(1,5),(2,7),(3,8),(4,9),(6,10)]
=> [5,7,8,9,1,10,2,3,4,6] => [5,1,7,2,8,3,9,4,10,6] => ? = 0
[(1,6),(2,7),(3,8),(4,9),(5,10)]
=> [6,7,8,9,10,1,2,3,4,5] => [6,1,7,2,8,3,9,4,10,5] => ? = 0
[(1,5),(2,6),(3,8),(4,9),(7,10)]
=> [5,6,8,9,1,2,10,3,4,7] => [5,1,6,2,8,3,9,4,10,7] => ? = 0
[(1,4),(2,6),(3,8),(5,9),(7,10)]
=> [4,6,8,1,9,2,10,3,5,7] => [4,1,6,2,8,3,9,5,10,7] => ? = 0
[(1,3),(2,6),(4,8),(5,9),(7,10)]
=> [3,6,1,8,9,2,10,4,5,7] => [3,1,6,2,8,4,9,5,10,7] => ? = 0
[(1,2),(3,6),(4,8),(5,9),(7,10)]
=> [2,1,6,8,9,3,10,4,5,7] => [2,1,6,3,8,4,9,5,10,7] => ? = 0
[(1,2),(3,5),(4,8),(6,9),(7,10)]
=> [2,1,5,8,3,9,10,4,6,7] => [2,1,5,3,8,4,9,6,10,7] => ? = 0
Description
The number of occurrences of the pattern 23-1.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern 23\!\!-\!\!1.
Matching statistic: St000223
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Mp00066: Permutations —inverse⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 29%
Values
[(1,2)]
=> [2,1] => [2,1] => 0
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)]
=> [3,4,1,2] => [3,4,1,2] => 0
[(1,4),(2,3)]
=> [3,4,2,1] => [4,3,1,2] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [3,4,1,2,6,5] => 0
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,3,1,2,6,5] => 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [5,3,1,6,2,4] => 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,3,1,5,2,4] => 2
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6,4,5,1,2,3] => 2
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [5,4,6,1,2,3] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [3,5,1,6,2,4] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,5,6,3,4] => 0
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,3,4] => 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [3,6,1,5,2,4] => 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [4,6,5,1,2,3] => 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [5,6,4,1,2,3] => 2
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [6,5,4,1,2,3] => 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] => 0
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [3,4,1,2,6,5,8,7] => 0
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,1,2,6,5,8,7] => ? = 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [5,3,1,6,2,4,8,7] => ? = 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [6,3,1,5,2,4,8,7] => 2
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [7,3,1,5,2,8,4,6] => ? = 2
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [8,3,1,5,2,7,4,6] => ? = 3
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [8,4,5,1,2,7,3,6] => ? = 3
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [7,4,5,1,2,8,3,6] => ? = 2
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [6,4,5,1,2,3,8,7] => ? = 2
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [5,4,6,1,2,3,8,7] => ? = 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [4,5,6,1,2,3,8,7] => 0
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [3,5,1,6,2,4,8,7] => 0
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,5,6,3,4,8,7] => 0
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,6,5,3,4,8,7] => ? = 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [3,6,1,5,2,4,8,7] => ? = 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [4,6,5,1,2,3,8,7] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [5,6,4,1,2,3,8,7] => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [6,5,4,1,2,3,8,7] => ? = 3
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [7,5,4,1,2,8,3,6] => ? = 3
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [8,5,4,1,2,7,3,6] => ? = 4
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [8,6,4,1,7,2,3,5] => ? = 4
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [7,6,4,1,8,2,3,5] => ? = 3
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [6,7,4,1,8,2,3,5] => ? = 2
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [5,7,4,1,2,8,3,6] => ? = 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [4,7,5,1,2,8,3,6] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [3,7,1,5,2,8,4,6] => ? = 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [2,1,7,5,3,8,4,6] => ? = 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,5,3,7,4,6] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [3,8,1,5,2,7,4,6] => ? = 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [4,8,5,1,2,7,3,6] => ? = 2
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [5,8,4,1,2,7,3,6] => ? = 3
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [6,8,4,1,7,2,3,5] => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [7,8,4,1,6,2,3,5] => 4
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [8,7,4,1,6,2,3,5] => ? = 5
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [8,7,5,6,1,2,3,4] => 5
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [7,8,5,6,1,2,3,4] => 4
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [6,8,5,7,1,2,3,4] => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [5,8,6,7,1,2,3,4] => ? = 2
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [4,8,6,1,7,2,3,5] => ? = 2
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [3,8,1,6,7,2,4,5] => ? = 2
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [2,1,8,6,7,3,4,5] => ? = 2
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [2,1,7,6,8,3,4,5] => ? = 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [3,7,1,6,8,2,4,5] => ? = 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [4,7,6,1,8,2,3,5] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [5,7,6,8,1,2,3,4] => ? = 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [6,7,5,8,1,2,3,4] => ? = 2
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [7,6,5,8,1,2,3,4] => ? = 3
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [8,6,5,7,1,2,3,4] => ? = 4
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [8,5,6,7,1,2,3,4] => 3
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [7,5,6,8,1,2,3,4] => ? = 2
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [6,5,7,8,1,2,3,4] => ? = 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [5,6,7,8,1,2,3,4] => 0
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [4,6,7,1,8,2,3,5] => 0
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [3,6,1,7,8,2,4,5] => 0
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,6,7,8,3,4,5] => 0
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,1,5,7,3,8,4,6] => 0
[(1,3),(2,5),(4,7),(6,8)]
=> [3,5,1,7,2,8,4,6] => [3,5,1,7,2,8,4,6] => 0
[(1,4),(2,5),(3,7),(6,8)]
=> [4,5,7,1,2,8,3,6] => [4,5,7,1,2,8,3,6] => 0
[(1,5),(2,4),(3,7),(6,8)]
=> [4,5,7,2,1,8,3,6] => [5,4,7,1,2,8,3,6] => ? = 1
[(1,6),(2,4),(3,7),(5,8)]
=> [4,6,7,2,8,1,3,5] => [6,4,7,1,8,2,3,5] => ? = 1
[(1,7),(2,4),(3,6),(5,8)]
=> [4,6,7,2,8,3,1,5] => [7,4,6,1,8,2,3,5] => ? = 2
[(1,8),(2,4),(3,6),(5,7)]
=> [4,6,7,2,8,3,5,1] => [8,4,6,1,7,2,3,5] => ? = 3
[(1,8),(2,3),(4,6),(5,7)]
=> [3,6,2,7,8,4,5,1] => [8,3,1,6,7,2,4,5] => ? = 3
[(1,7),(2,3),(4,6),(5,8)]
=> [3,6,2,7,8,4,1,5] => [7,3,1,6,8,2,4,5] => ? = 2
[(1,6),(2,3),(4,7),(5,8)]
=> [3,6,2,7,8,1,4,5] => [6,3,1,7,8,2,4,5] => ? = 1
[(1,5),(2,3),(4,7),(6,8)]
=> [3,5,2,7,1,8,4,6] => [5,3,1,7,2,8,4,6] => ? = 1
[(1,4),(2,3),(5,7),(6,8)]
=> [3,4,2,1,7,8,5,6] => [4,3,1,2,7,8,5,6] => ? = 1
[(1,3),(2,4),(5,7),(6,8)]
=> [3,4,1,2,7,8,5,6] => [3,4,1,2,7,8,5,6] => 0
[(1,2),(3,4),(5,7),(6,8)]
=> [2,1,4,3,7,8,5,6] => [2,1,4,3,7,8,5,6] => 0
[(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [4,3,1,2,8,7,5,6] => 2
[(1,5),(2,3),(4,8),(6,7)]
=> [3,5,2,7,1,8,6,4] => [5,3,1,8,2,7,4,6] => 2
[(1,6),(2,4),(3,8),(5,7)]
=> [4,6,7,2,8,1,5,3] => [6,4,8,1,7,2,3,5] => 2
[(1,6),(2,5),(3,8),(4,7)]
=> [5,6,7,8,2,1,4,3] => [6,5,8,7,1,2,3,4] => 2
[(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => 6
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 0
[(1,6),(2,7),(3,8),(4,9),(5,10)]
=> [6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,10,1,2,3,4,5] => 0
[(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,1,2,3,4,5] => 10
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 0
[(1,7),(2,8),(3,9),(4,10),(5,11),(6,12)]
=> [7,8,9,10,11,12,1,2,3,4,5,6] => [7,8,9,10,11,12,1,2,3,4,5,6] => 0
Description
The number of nestings in the permutation.
Matching statistic: St000356
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 25%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 25%
Values
[(1,2)]
=> [2,1] => [2,1] => [1,2] => 0
[(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[(1,3),(2,4)]
=> [3,4,1,2] => [4,1,3,2] => [2,3,1,4] => 0
[(1,4),(2,3)]
=> [3,4,2,1] => [4,2,3,1] => [1,3,2,4] => 1
[(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => 0
[(1,3),(2,4),(5,6)]
=> [3,4,1,2,6,5] => [4,1,3,2,6,5] => [5,6,2,3,1,4] => 0
[(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,2,3,1,6,5] => [5,6,1,3,2,4] => 1
[(1,5),(2,3),(4,6)]
=> [3,5,2,6,1,4] => [6,2,3,1,5,4] => [4,5,1,3,2,6] => 1
[(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => 2
[(1,6),(2,4),(3,5)]
=> [4,5,6,2,3,1] => [6,2,5,3,4,1] => [1,4,3,5,2,6] => 2
[(1,5),(2,4),(3,6)]
=> [4,5,6,2,1,3] => [6,2,5,1,4,3] => [3,4,1,5,2,6] => 1
[(1,4),(2,5),(3,6)]
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => [3,4,2,5,1,6] => 0
[(1,3),(2,5),(4,6)]
=> [3,5,1,6,2,4] => [6,1,3,2,5,4] => [4,5,2,3,1,6] => 0
[(1,2),(3,5),(4,6)]
=> [2,1,5,6,3,4] => [2,1,6,3,5,4] => [4,5,3,6,1,2] => 0
[(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,4,5,3] => [3,5,4,6,1,2] => 1
[(1,3),(2,6),(4,5)]
=> [3,5,1,6,4,2] => [6,4,5,1,3,2] => [2,3,1,5,4,6] => 1
[(1,4),(2,6),(3,5)]
=> [4,5,6,1,3,2] => [6,1,5,3,4,2] => [2,4,3,5,1,6] => 1
[(1,5),(2,6),(3,4)]
=> [4,5,6,3,1,2] => [6,3,5,1,4,2] => [2,4,1,5,3,6] => 2
[(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [6,3,5,2,4,1] => [1,4,2,5,3,6] => 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] => [7,8,5,6,3,4,1,2] => 0
[(1,3),(2,4),(5,6),(7,8)]
=> [3,4,1,2,6,5,8,7] => [4,1,3,2,6,5,8,7] => [7,8,5,6,2,3,1,4] => ? = 0
[(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,2,3,1,6,5,8,7] => [7,8,5,6,1,3,2,4] => ? = 1
[(1,5),(2,3),(4,6),(7,8)]
=> [3,5,2,6,1,4,8,7] => [6,2,3,1,5,4,8,7] => [7,8,4,5,1,3,2,6] => ? = 1
[(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [6,4,5,2,3,1,8,7] => [7,8,1,3,2,5,4,6] => 2
[(1,7),(2,3),(4,5),(6,8)]
=> [3,5,2,7,4,8,1,6] => [8,4,5,2,3,1,7,6] => [6,7,1,3,2,5,4,8] => ? = 2
[(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [8,6,7,4,5,2,3,1] => [1,3,2,5,4,7,6,8] => 3
[(1,8),(2,4),(3,5),(6,7)]
=> [4,5,7,2,3,8,6,1] => [8,6,7,2,5,3,4,1] => [1,4,3,5,2,7,6,8] => ? = 3
[(1,7),(2,4),(3,5),(6,8)]
=> [4,5,7,2,3,8,1,6] => [8,2,5,3,4,1,7,6] => [6,7,1,4,3,5,2,8] => ? = 2
[(1,6),(2,4),(3,5),(7,8)]
=> [4,5,6,2,3,1,8,7] => [6,2,5,3,4,1,8,7] => [7,8,1,4,3,5,2,6] => ? = 2
[(1,5),(2,4),(3,6),(7,8)]
=> [4,5,6,2,1,3,8,7] => [6,2,5,1,4,3,8,7] => [7,8,3,4,1,5,2,6] => ? = 1
[(1,4),(2,5),(3,6),(7,8)]
=> [4,5,6,1,2,3,8,7] => [6,1,5,2,4,3,8,7] => [7,8,3,4,2,5,1,6] => ? = 0
[(1,3),(2,5),(4,6),(7,8)]
=> [3,5,1,6,2,4,8,7] => [6,1,3,2,5,4,8,7] => [7,8,4,5,2,3,1,6] => ? = 0
[(1,2),(3,5),(4,6),(7,8)]
=> [2,1,5,6,3,4,8,7] => [2,1,6,3,5,4,8,7] => [7,8,4,5,3,6,1,2] => ? = 0
[(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,6,4,5,3,8,7] => [7,8,3,5,4,6,1,2] => ? = 1
[(1,3),(2,6),(4,5),(7,8)]
=> [3,5,1,6,4,2,8,7] => [6,4,5,1,3,2,8,7] => [7,8,2,3,1,5,4,6] => ? = 1
[(1,4),(2,6),(3,5),(7,8)]
=> [4,5,6,1,3,2,8,7] => [6,1,5,3,4,2,8,7] => [7,8,2,4,3,5,1,6] => ? = 1
[(1,5),(2,6),(3,4),(7,8)]
=> [4,5,6,3,1,2,8,7] => [6,3,5,1,4,2,8,7] => [7,8,2,4,1,5,3,6] => ? = 2
[(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [6,3,5,2,4,1,8,7] => [7,8,1,4,2,5,3,6] => ? = 3
[(1,7),(2,5),(3,4),(6,8)]
=> [4,5,7,3,2,8,1,6] => [8,3,5,2,4,1,7,6] => [6,7,1,4,2,5,3,8] => ? = 3
[(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [8,6,7,3,5,2,4,1] => [1,4,2,5,3,7,6,8] => 4
[(1,8),(2,6),(3,4),(5,7)]
=> [4,6,7,3,8,2,5,1] => [8,3,7,5,6,2,4,1] => [1,4,2,6,5,7,3,8] => ? = 4
[(1,7),(2,6),(3,4),(5,8)]
=> [4,6,7,3,8,2,1,5] => [8,3,7,2,4,1,6,5] => [5,6,1,4,2,7,3,8] => ? = 3
[(1,6),(2,7),(3,4),(5,8)]
=> [4,6,7,3,8,1,2,5] => [8,3,7,1,4,2,6,5] => [5,6,2,4,1,7,3,8] => ? = 2
[(1,5),(2,7),(3,4),(6,8)]
=> [4,5,7,3,1,8,2,6] => [8,3,5,1,4,2,7,6] => [6,7,2,4,1,5,3,8] => ? = 2
[(1,4),(2,7),(3,5),(6,8)]
=> [4,5,7,1,3,8,2,6] => [8,1,5,3,4,2,7,6] => [6,7,2,4,3,5,1,8] => ? = 1
[(1,3),(2,7),(4,5),(6,8)]
=> [3,5,1,7,4,8,2,6] => [8,4,5,1,3,2,7,6] => [6,7,2,3,1,5,4,8] => ? = 1
[(1,2),(3,7),(4,5),(6,8)]
=> [2,1,5,7,4,8,3,6] => [2,1,8,4,5,3,7,6] => [6,7,3,5,4,8,1,2] => ? = 1
[(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,6,7,4,5,3] => [3,5,4,7,6,8,1,2] => 2
[(1,3),(2,8),(4,5),(6,7)]
=> [3,5,1,7,4,8,6,2] => [8,6,7,4,5,1,3,2] => [2,3,1,5,4,7,6,8] => ? = 2
[(1,4),(2,8),(3,5),(6,7)]
=> [4,5,7,1,3,8,6,2] => [8,6,7,1,5,3,4,2] => [2,4,3,5,1,7,6,8] => ? = 2
[(1,5),(2,8),(3,4),(6,7)]
=> [4,5,7,3,1,8,6,2] => [8,6,7,3,5,1,4,2] => [2,4,1,5,3,7,6,8] => ? = 3
[(1,6),(2,8),(3,4),(5,7)]
=> [4,6,7,3,8,1,5,2] => [8,3,7,5,6,1,4,2] => [2,4,1,6,5,7,3,8] => ? = 3
[(1,7),(2,8),(3,4),(5,6)]
=> [4,6,7,3,8,5,1,2] => [8,5,7,3,6,1,4,2] => [2,4,1,6,3,7,5,8] => ? = 4
[(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [8,5,7,3,6,2,4,1] => [1,4,2,6,3,7,5,8] => 5
[(1,8),(2,7),(3,5),(4,6)]
=> [5,6,7,8,3,4,2,1] => [8,3,7,4,6,2,5,1] => [1,5,2,6,4,7,3,8] => ? = 5
[(1,7),(2,8),(3,5),(4,6)]
=> [5,6,7,8,3,4,1,2] => [8,3,7,4,6,1,5,2] => [2,5,1,6,4,7,3,8] => ? = 4
[(1,6),(2,8),(3,5),(4,7)]
=> [5,6,7,8,3,1,4,2] => [8,3,7,1,6,4,5,2] => [2,5,4,6,1,7,3,8] => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
=> [5,6,7,8,1,3,4,2] => [8,1,7,3,6,4,5,2] => [2,5,4,6,3,7,1,8] => ? = 2
[(1,4),(2,8),(3,6),(5,7)]
=> [4,6,7,1,8,3,5,2] => [8,1,7,5,6,3,4,2] => [2,4,3,6,5,7,1,8] => ? = 2
[(1,3),(2,8),(4,6),(5,7)]
=> [3,6,1,7,8,4,5,2] => [8,4,7,5,6,1,3,2] => [2,3,1,6,5,7,4,8] => ? = 2
[(1,2),(3,8),(4,6),(5,7)]
=> [2,1,6,7,8,4,5,3] => [2,1,8,4,7,5,6,3] => [3,6,5,7,4,8,1,2] => ? = 2
[(1,2),(3,7),(4,6),(5,8)]
=> [2,1,6,7,8,4,3,5] => [2,1,8,4,7,3,6,5] => [5,6,3,7,4,8,1,2] => 1
[(1,3),(2,7),(4,6),(5,8)]
=> [3,6,1,7,8,4,2,5] => [8,4,7,1,3,2,6,5] => [5,6,2,3,1,7,4,8] => ? = 1
[(1,4),(2,7),(3,6),(5,8)]
=> [4,6,7,1,8,3,2,5] => [8,1,7,3,4,2,6,5] => [5,6,2,4,3,7,1,8] => ? = 1
[(1,5),(2,7),(3,6),(4,8)]
=> [5,6,7,8,1,3,2,4] => [8,1,7,3,6,2,5,4] => [4,5,2,6,3,7,1,8] => ? = 1
[(1,6),(2,7),(3,5),(4,8)]
=> [5,6,7,8,3,1,2,4] => [8,3,7,1,6,2,5,4] => [4,5,2,6,1,7,3,8] => ? = 2
[(1,7),(2,6),(3,5),(4,8)]
=> [5,6,7,8,3,2,1,4] => [8,3,7,2,6,1,5,4] => [4,5,1,6,2,7,3,8] => ? = 3
[(1,8),(2,6),(3,5),(4,7)]
=> [5,6,7,8,3,2,4,1] => [8,3,7,2,6,4,5,1] => [1,5,4,6,2,7,3,8] => ? = 4
[(1,8),(2,5),(3,6),(4,7)]
=> [5,6,7,8,2,3,4,1] => [8,2,7,3,6,4,5,1] => [1,5,4,6,3,7,2,8] => ? = 3
[(1,7),(2,5),(3,6),(4,8)]
=> [5,6,7,8,2,3,1,4] => [8,2,7,3,6,1,5,4] => [4,5,1,6,3,7,2,8] => ? = 2
[(1,6),(2,5),(3,7),(4,8)]
=> [5,6,7,8,2,1,3,4] => [8,2,7,1,6,3,5,4] => [4,5,3,6,1,7,2,8] => ? = 1
[(1,5),(2,6),(3,7),(4,8)]
=> [5,6,7,8,1,2,3,4] => [8,1,7,2,6,3,5,4] => [4,5,3,6,2,7,1,8] => ? = 0
[(1,4),(2,6),(3,7),(5,8)]
=> [4,6,7,1,8,2,3,5] => [8,1,7,2,4,3,6,5] => [5,6,3,4,2,7,1,8] => ? = 0
[(1,3),(2,6),(4,7),(5,8)]
=> [3,6,1,7,8,2,4,5] => [8,1,3,2,7,4,6,5] => [5,6,4,7,2,3,1,8] => ? = 0
[(1,2),(3,6),(4,7),(5,8)]
=> [2,1,6,7,8,3,4,5] => [2,1,8,3,7,4,6,5] => [5,6,4,7,3,8,1,2] => ? = 0
[(1,2),(3,5),(4,7),(6,8)]
=> [2,1,5,7,3,8,4,6] => [2,1,8,3,5,4,7,6] => [6,7,4,5,3,8,1,2] => ? = 0
[(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [4,2,3,1,8,6,7,5] => [5,7,6,8,1,3,2,4] => 2
[(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [8,5,7,4,6,2,3,1] => [1,3,2,6,4,7,5,8] => 4
[(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [8,4,7,3,6,2,5,1] => [1,5,2,6,3,7,4,8] => 6
[(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => [9,10,7,8,5,6,3,4,1,2] => 0
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern 13\!\!-\!\!2.
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