- FindStat

Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000042
(load all 13 compositions to match this statistic)

Mapping

St000042: 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)]
 => 1
[(1,4),(2,3)]
 => 0
[(1,2),(3,4),(5,6)]
 => 0
[(1,3),(2,4),(5,6)]
 => 1
[(1,4),(2,3),(5,6)]
 => 0
[(1,5),(2,3),(4,6)]
 => 1
[(1,6),(2,3),(4,5)]
 => 0
[(1,6),(2,4),(3,5)]
 => 1
[(1,5),(2,4),(3,6)]
 => 2
[(1,4),(2,5),(3,6)]
 => 3
[(1,3),(2,5),(4,6)]
 => 2
[(1,2),(3,5),(4,6)]
 => 1
[(1,2),(3,6),(4,5)]
 => 0
[(1,3),(2,6),(4,5)]
 => 1
[(1,4),(2,6),(3,5)]
 => 2
[(1,5),(2,6),(3,4)]
 => 1
[(1,6),(2,5),(3,4)]
 => 0
[(1,2),(3,4),(5,6),(7,8)]
 => 0
[(1,3),(2,4),(5,6),(7,8)]
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => 0
[(1,5),(2,3),(4,6),(7,8)]
 => 1
[(1,6),(2,3),(4,5),(7,8)]
 => 0
[(1,7),(2,3),(4,5),(6,8)]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => 0
[(1,8),(2,4),(3,5),(6,7)]
 => 1
[(1,7),(2,4),(3,5),(6,8)]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => 2
[(1,4),(2,5),(3,6),(7,8)]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => 0
[(1,3),(2,6),(4,5),(7,8)]
 => 1
[(1,4),(2,6),(3,5),(7,8)]
 => 2
[(1,5),(2,6),(3,4),(7,8)]
 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => 0
[(1,7),(2,5),(3,4),(6,8)]
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => 0
[(1,8),(2,6),(3,4),(5,7)]
 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => 2
[(1,6),(2,7),(3,4),(5,8)]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => 2
[(1,2),(3,7),(4,5),(6,8)]
 => 1
[(1,2),(3,8),(4,5),(6,7)]
 => 0
[(1,3),(2,8),(4,5),(6,7)]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => 2

Description

The number of crossings of a perfect matching. This is the number of pairs of edges

$((a,b),(c,d))$ such that

$a\le c\le b\le d$ , 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

$\{1,\dots,2n\}$ according to the number of crossings

$\textrm{cr}(M)$ 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: St000041
(load all 12 compositions to match this statistic)

Mapping

Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 75%

Values

[(1,2)]
 => [(1,2)]
 => 0
[(1,2),(3,4)]
 => [(1,2),(3,4)]
 => 0
[(1,3),(2,4)]
 => [(1,4),(2,3)]
 => 1
[(1,4),(2,3)]
 => [(1,3),(2,4)]
 => 0
[(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)]
 => 1
[(1,4),(2,3),(5,6)]
 => [(1,3),(2,4),(5,6)]
 => 0
[(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)]
 => 0
[(1,6),(2,4),(3,5)]
 => [(1,5),(2,4),(3,6)]
 => 1
[(1,5),(2,4),(3,6)]
 => [(1,6),(2,4),(3,5)]
 => 2
[(1,4),(2,5),(3,6)]
 => [(1,6),(2,5),(3,4)]
 => 3
[(1,3),(2,5),(4,6)]
 => [(1,6),(2,3),(4,5)]
 => 2
[(1,2),(3,5),(4,6)]
 => [(1,2),(3,6),(4,5)]
 => 1
[(1,2),(3,6),(4,5)]
 => [(1,2),(3,5),(4,6)]
 => 0
[(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)]
 => 2
[(1,5),(2,6),(3,4)]
 => [(1,4),(2,6),(3,5)]
 => 1
[(1,6),(2,5),(3,4)]
 => [(1,4),(2,5),(3,6)]
 => 0
[(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)]
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => [(1,3),(2,4),(5,6),(7,8)]
 => 0
[(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)]
 => 0
[(1,7),(2,3),(4,5),(6,8)]
 => [(1,3),(2,5),(4,8),(6,7)]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [(1,3),(2,5),(4,7),(6,8)]
 => 0
[(1,8),(2,4),(3,5),(6,7)]
 => [(1,5),(2,4),(3,7),(6,8)]
 => 1
[(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)]
 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => [(1,6),(2,4),(3,5),(7,8)]
 => 2
[(1,4),(2,5),(3,6),(7,8)]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => [(1,2),(3,5),(4,6),(7,8)]
 => 0
[(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)]
 => 2
[(1,5),(2,6),(3,4),(7,8)]
 => [(1,4),(2,6),(3,5),(7,8)]
 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => [(1,4),(2,5),(3,6),(7,8)]
 => 0
[(1,7),(2,5),(3,4),(6,8)]
 => [(1,4),(2,5),(3,8),(6,7)]
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [(1,4),(2,5),(3,7),(6,8)]
 => 0
[(1,8),(2,6),(3,4),(5,7)]
 => [(1,4),(2,7),(3,6),(5,8)]
 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => [(1,4),(2,8),(3,6),(5,7)]
 => 2
[(1,6),(2,7),(3,4),(5,8)]
 => [(1,4),(2,8),(3,7),(5,6)]
 => 3
[(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)]
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [(1,5),(2,3),(4,8),(6,7)]
 => 2
[(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)]
 => 0
[(1,3),(2,8),(4,5),(6,7)]
 => [(1,5),(2,3),(4,7),(6,8)]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [(1,5),(2,7),(3,4),(6,8)]
 => 2
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [(1,3),(2,6),(4,5),(7,8),(9,10)]
 => ? = 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [(1,3),(2,5),(4,8),(6,7),(9,10)]
 => ? = 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => [(1,3),(2,5),(4,7),(6,10),(8,9)]
 => ? = 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
 => [(1,5),(2,4),(3,7),(6,9),(8,10)]
 => ? = 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
 => [(1,5),(2,4),(3,7),(6,10),(8,9)]
 => ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [(1,5),(2,4),(3,7),(6,8),(9,10)]
 => ? = 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [(1,5),(2,4),(3,8),(6,7),(9,10)]
 => ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [(1,5),(2,4),(3,6),(7,8),(9,10)]
 => ? = 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [(1,6),(2,4),(3,5),(7,8),(9,10)]
 => ? = 2
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [(1,5),(2,3),(4,6),(7,8),(9,10)]
 => ? = 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [(1,5),(2,6),(3,4),(7,8),(9,10)]
 => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => [(1,4),(2,6),(3,5),(7,8),(9,10)]
 => ? = 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
 => [(1,4),(2,5),(3,8),(6,7),(9,10)]
 => ? = 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
 => [(1,4),(2,5),(3,7),(6,10),(8,9)]
 => ? = 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
 => [(1,4),(2,7),(3,6),(5,9),(8,10)]
 => ? = 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
 => [(1,4),(2,7),(3,6),(5,10),(8,9)]
 => ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
 => [(1,4),(2,7),(3,6),(5,8),(9,10)]
 => ? = 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
 => [(1,4),(2,8),(3,6),(5,7),(9,10)]
 => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
 => [(1,4),(2,8),(3,7),(5,6),(9,10)]
 => ? = 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
 => [(1,4),(2,8),(3,5),(6,7),(9,10)]
 => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
 => [(1,5),(2,8),(3,4),(6,7),(9,10)]
 => ? = 3
[(1,3),(2,7),(4,5),(6,8),(9,10)]
 => [(1,5),(2,3),(4,8),(6,7),(9,10)]
 => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
 => [(1,2),(3,5),(4,8),(6,7),(9,10)]
 => ? = 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
 => [(1,5),(2,3),(4,7),(6,8),(9,10)]
 => ? = 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
 => [(1,5),(2,7),(3,4),(6,8),(9,10)]
 => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
 => [(1,4),(2,7),(3,5),(6,8),(9,10)]
 => ? = 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
 => [(1,4),(2,7),(3,8),(5,6),(9,10)]
 => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
 => [(1,4),(2,6),(3,8),(5,7),(9,10)]
 => ? = 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
 => [(1,4),(2,6),(3,7),(5,10),(8,9)]
 => ? = 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
 => [(1,4),(2,6),(3,9),(5,8),(7,10)]
 => ? = 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
 => [(1,4),(2,6),(3,10),(5,8),(7,9)]
 => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
 => [(1,4),(2,6),(3,10),(5,9),(7,8)]
 => ? = 3
[(1,7),(2,9),(3,4),(5,6),(8,10)]
 => [(1,4),(2,6),(3,10),(5,7),(8,9)]
 => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
 => [(1,4),(2,7),(3,10),(5,6),(8,9)]
 => ? = 3
[(1,5),(2,9),(3,4),(6,7),(8,10)]
 => [(1,4),(2,7),(3,5),(6,10),(8,9)]
 => ? = 2
[(1,4),(2,9),(3,5),(6,7),(8,10)]
 => [(1,5),(2,7),(3,4),(6,10),(8,9)]
 => ? = 3
[(1,3),(2,9),(4,5),(6,7),(8,10)]
 => [(1,5),(2,3),(4,7),(6,10),(8,9)]
 => ? = 2
[(1,2),(3,9),(4,5),(6,7),(8,10)]
 => [(1,2),(3,5),(4,7),(6,10),(8,9)]
 => ? = 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
 => [(1,5),(2,3),(4,7),(6,9),(8,10)]
 => ? = 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
 => [(1,5),(2,7),(3,4),(6,9),(8,10)]
 => ? = 2
[(1,5),(2,10),(3,4),(6,7),(8,9)]
 => [(1,4),(2,7),(3,5),(6,9),(8,10)]
 => ? = 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
 => [(1,4),(2,7),(3,9),(5,6),(8,10)]
 => ? = 2
[(1,7),(2,10),(3,4),(5,6),(8,9)]
 => [(1,4),(2,6),(3,9),(5,7),(8,10)]
 => ? = 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
 => [(1,4),(2,6),(3,9),(5,10),(7,8)]
 => ? = 2
[(1,9),(2,10),(3,4),(5,6),(7,8)]
 => [(1,4),(2,6),(3,8),(5,10),(7,9)]
 => ? = 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
 => [(1,6),(2,5),(3,8),(4,9),(7,10)]
 => ? = 1
[(1,9),(2,10),(3,5),(4,6),(7,8)]
 => [(1,6),(2,5),(3,8),(4,10),(7,9)]
 => ? = 2
[(1,8),(2,10),(3,5),(4,6),(7,9)]
 => [(1,6),(2,5),(3,9),(4,10),(7,8)]
 => ? = 3
[(1,7),(2,10),(3,5),(4,6),(8,9)]
 => [(1,6),(2,5),(3,9),(4,7),(8,10)]
 => ? = 2
[(1,6),(2,10),(3,5),(4,7),(8,9)]
 => [(1,7),(2,5),(3,9),(4,6),(8,10)]
 => ? = 3

Description

The number of nestings of a perfect matching. This is the number of pairs of edges

$((a,b), (c,d))$ such that

$a\le c\le d\le b$ . i.e., the edge

$(c,d)$ is nested inside

$(a,b)$ .

Matching statistic: St000232
(load all 9 compositions to match this statistic)

Mapping

Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 44%

Values

[(1,2)]
 => {{1,2}}
 => 0
[(1,2),(3,4)]
 => {{1,2},{3,4}}
 => 0
[(1,3),(2,4)]
 => {{1,3},{2,4}}
 => 1
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => 0
[(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
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => 0
[(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}}
 => 0
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => 2
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => 3
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => 2
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => 0
[(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}}
 => 2
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => 0
[(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
[(1,4),(2,3),(5,6),(7,8)]
 => {{1,4},{2,3},{5,6},{7,8}}
 => 0
[(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}}
 => 0
[(1,7),(2,3),(4,5),(6,8)]
 => {{1,7},{2,3},{4,5},{6,8}}
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => {{1,8},{2,3},{4,5},{6,7}}
 => 0
[(1,8),(2,4),(3,5),(6,7)]
 => {{1,8},{2,4},{3,5},{6,7}}
 => 1
[(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}}
 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => {{1,5},{2,4},{3,6},{7,8}}
 => 2
[(1,4),(2,5),(3,6),(7,8)]
 => {{1,4},{2,5},{3,6},{7,8}}
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => {{1,3},{2,5},{4,6},{7,8}}
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => {{1,2},{3,5},{4,6},{7,8}}
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => {{1,2},{3,6},{4,5},{7,8}}
 => 0
[(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}}
 => 2
[(1,5),(2,6),(3,4),(7,8)]
 => {{1,5},{2,6},{3,4},{7,8}}
 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => {{1,6},{2,5},{3,4},{7,8}}
 => 0
[(1,7),(2,5),(3,4),(6,8)]
 => {{1,7},{2,5},{3,4},{6,8}}
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => {{1,8},{2,5},{3,4},{6,7}}
 => 0
[(1,8),(2,6),(3,4),(5,7)]
 => {{1,8},{2,6},{3,4},{5,7}}
 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => {{1,7},{2,6},{3,4},{5,8}}
 => 2
[(1,6),(2,7),(3,4),(5,8)]
 => {{1,6},{2,7},{3,4},{5,8}}
 => 3
[(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}}
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => {{1,3},{2,7},{4,5},{6,8}}
 => 2
[(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}}
 => 0
[(1,3),(2,8),(4,5),(6,7)]
 => {{1,3},{2,8},{4,5},{6,7}}
 => 1
[(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}}
 => ? = 1
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => {{1,5},{2,3},{4,6},{7,8},{9,10}}
 => ? = 1
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => {{1,7},{2,3},{4,5},{6,8},{9,10}}
 => ? = 1
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => {{1,9},{2,3},{4,5},{6,7},{8,10}}
 => ? = 1
[(1,10),(2,4),(3,5),(6,7),(8,9)]
 => {{1,10},{2,4},{3,5},{6,7},{8,9}}
 => ? = 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
 => {{1,9},{2,4},{3,5},{6,7},{8,10}}
 => ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => {{1,8},{2,4},{3,5},{6,7},{9,10}}
 => ? = 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => {{1,7},{2,4},{3,5},{6,8},{9,10}}
 => ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => {{1,6},{2,4},{3,5},{7,8},{9,10}}
 => ? = 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => {{1,5},{2,4},{3,6},{7,8},{9,10}}
 => ? = 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => {{1,4},{2,5},{3,6},{7,8},{9,10}}
 => ? = 3
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => {{1,3},{2,5},{4,6},{7,8},{9,10}}
 => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => {{1,2},{3,5},{4,6},{7,8},{9,10}}
 => ? = 1
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => {{1,3},{2,6},{4,5},{7,8},{9,10}}
 => ? = 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => {{1,4},{2,6},{3,5},{7,8},{9,10}}
 => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => {{1,5},{2,6},{3,4},{7,8},{9,10}}
 => ? = 1
[(1,7),(2,5),(3,4),(6,8),(9,10)]
 => {{1,7},{2,5},{3,4},{6,8},{9,10}}
 => ? = 1
[(1,9),(2,5),(3,4),(6,7),(8,10)]
 => {{1,9},{2,5},{3,4},{6,7},{8,10}}
 => ? = 1
[(1,10),(2,6),(3,4),(5,7),(8,9)]
 => {{1,10},{2,6},{3,4},{5,7},{8,9}}
 => ? = 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
 => {{1,9},{2,6},{3,4},{5,7},{8,10}}
 => ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
 => {{1,8},{2,6},{3,4},{5,7},{9,10}}
 => ? = 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
 => {{1,7},{2,6},{3,4},{5,8},{9,10}}
 => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
 => {{1,6},{2,7},{3,4},{5,8},{9,10}}
 => ? = 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
 => {{1,5},{2,7},{3,4},{6,8},{9,10}}
 => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
 => {{1,4},{2,7},{3,5},{6,8},{9,10}}
 => ? = 3
[(1,3),(2,7),(4,5),(6,8),(9,10)]
 => {{1,3},{2,7},{4,5},{6,8},{9,10}}
 => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
 => {{1,2},{3,7},{4,5},{6,8},{9,10}}
 => ? = 1
[(1,3),(2,8),(4,5),(6,7),(9,10)]
 => {{1,3},{2,8},{4,5},{6,7},{9,10}}
 => ? = 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
 => {{1,4},{2,8},{3,5},{6,7},{9,10}}
 => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
 => {{1,5},{2,8},{3,4},{6,7},{9,10}}
 => ? = 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
 => {{1,6},{2,8},{3,4},{5,7},{9,10}}
 => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
 => {{1,7},{2,8},{3,4},{5,6},{9,10}}
 => ? = 1
[(1,9),(2,7),(3,4),(5,6),(8,10)]
 => {{1,9},{2,7},{3,4},{5,6},{8,10}}
 => ? = 1
[(1,10),(2,8),(3,4),(5,6),(7,9)]
 => {{1,10},{2,8},{3,4},{5,6},{7,9}}
 => ? = 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
 => {{1,9},{2,8},{3,4},{5,6},{7,10}}
 => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
 => {{1,8},{2,9},{3,4},{5,6},{7,10}}
 => ? = 3
[(1,7),(2,9),(3,4),(5,6),(8,10)]
 => {{1,7},{2,9},{3,4},{5,6},{8,10}}
 => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
 => {{1,6},{2,9},{3,4},{5,7},{8,10}}
 => ? = 3
[(1,5),(2,9),(3,4),(6,7),(8,10)]
 => {{1,5},{2,9},{3,4},{6,7},{8,10}}
 => ? = 2
[(1,4),(2,9),(3,5),(6,7),(8,10)]
 => {{1,4},{2,9},{3,5},{6,7},{8,10}}
 => ? = 3
[(1,3),(2,9),(4,5),(6,7),(8,10)]
 => {{1,3},{2,9},{4,5},{6,7},{8,10}}
 => ? = 2
[(1,2),(3,9),(4,5),(6,7),(8,10)]
 => {{1,2},{3,9},{4,5},{6,7},{8,10}}
 => ? = 1
[(1,3),(2,10),(4,5),(6,7),(8,9)]
 => {{1,3},{2,10},{4,5},{6,7},{8,9}}
 => ? = 1
[(1,4),(2,10),(3,5),(6,7),(8,9)]
 => {{1,4},{2,10},{3,5},{6,7},{8,9}}
 => ? = 2
[(1,5),(2,10),(3,4),(6,7),(8,9)]
 => {{1,5},{2,10},{3,4},{6,7},{8,9}}
 => ? = 1
[(1,6),(2,10),(3,4),(5,7),(8,9)]
 => {{1,6},{2,10},{3,4},{5,7},{8,9}}
 => ? = 2
[(1,7),(2,10),(3,4),(5,6),(8,9)]
 => {{1,7},{2,10},{3,4},{5,6},{8,9}}
 => ? = 1
[(1,8),(2,10),(3,4),(5,6),(7,9)]
 => {{1,8},{2,10},{3,4},{5,6},{7,9}}
 => ? = 2
[(1,9),(2,10),(3,4),(5,6),(7,8)]
 => {{1,9},{2,10},{3,4},{5,6},{7,8}}
 => ? = 1
[(1,10),(2,9),(3,5),(4,6),(7,8)]
 => {{1,10},{2,9},{3,5},{4,6},{7,8}}
 => ? = 1

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: St000233
(load all 10 compositions to match this statistic)

Mapping

Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%

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}}
 => 1
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 0
[(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}}
 => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3},{2,4},{5,6}}
 => 0
[(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}}
 => 0
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5},{2,4},{3,6}}
 => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4},{3,5}}
 => 2
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => 3
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3},{4,5}}
 => 2
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2},{3,6},{4,5}}
 => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2},{3,5},{4,6}}
 => 0
[(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}}
 => 2
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4},{2,6},{3,5}}
 => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2,5},{3,6}}
 => 0
[(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}}
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => {{1,4},{2,3},{5,6},{7,8}}
 => {{1,3},{2,4},{5,6},{7,8}}
 => 0
[(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}}
 => 0
[(1,7),(2,3),(4,5),(6,8)]
 => {{1,7},{2,3},{4,5},{6,8}}
 => {{1,3},{2,5},{4,8},{6,7}}
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => {{1,8},{2,3},{4,5},{6,7}}
 => {{1,3},{2,5},{4,7},{6,8}}
 => 0
[(1,8),(2,4),(3,5),(6,7)]
 => {{1,8},{2,4},{3,5},{6,7}}
 => {{1,5},{2,4},{3,7},{6,8}}
 => 1
[(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}}
 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => {{1,5},{2,4},{3,6},{7,8}}
 => {{1,6},{2,4},{3,5},{7,8}}
 => 2
[(1,4),(2,5),(3,6),(7,8)]
 => {{1,4},{2,5},{3,6},{7,8}}
 => {{1,6},{2,5},{3,4},{7,8}}
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => {{1,3},{2,5},{4,6},{7,8}}
 => {{1,6},{2,3},{4,5},{7,8}}
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => {{1,2},{3,5},{4,6},{7,8}}
 => {{1,2},{3,6},{4,5},{7,8}}
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => {{1,2},{3,6},{4,5},{7,8}}
 => {{1,2},{3,5},{4,6},{7,8}}
 => 0
[(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}}
 => 2
[(1,5),(2,6),(3,4),(7,8)]
 => {{1,5},{2,6},{3,4},{7,8}}
 => {{1,4},{2,6},{3,5},{7,8}}
 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => {{1,6},{2,5},{3,4},{7,8}}
 => {{1,4},{2,5},{3,6},{7,8}}
 => 0
[(1,7),(2,5),(3,4),(6,8)]
 => {{1,7},{2,5},{3,4},{6,8}}
 => {{1,4},{2,5},{3,8},{6,7}}
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => {{1,8},{2,5},{3,4},{6,7}}
 => {{1,4},{2,5},{3,7},{6,8}}
 => 0
[(1,8),(2,6),(3,4),(5,7)]
 => {{1,8},{2,6},{3,4},{5,7}}
 => {{1,4},{2,7},{3,6},{5,8}}
 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => {{1,7},{2,6},{3,4},{5,8}}
 => {{1,4},{2,8},{3,6},{5,7}}
 => 2
[(1,6),(2,7),(3,4),(5,8)]
 => {{1,6},{2,7},{3,4},{5,8}}
 => {{1,4},{2,8},{3,7},{5,6}}
 => 3
[(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}}
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => {{1,3},{2,7},{4,5},{6,8}}
 => {{1,5},{2,3},{4,8},{6,7}}
 => 2
[(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}}
 => 0
[(1,3),(2,8),(4,5),(6,7)]
 => {{1,3},{2,8},{4,5},{6,7}}
 => {{1,5},{2,3},{4,7},{6,8}}
 => 1
[(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}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => {{1,5},{2,3},{4,6},{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}}
 => ? = 0
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => {{1,7},{2,3},{4,5},{6,8},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => {{1,9},{2,3},{4,5},{6,7},{8,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,10),(2,4),(3,5),(6,7),(8,9)]
 => {{1,10},{2,4},{3,5},{6,7},{8,9}}
 => ?
 => ? = 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
 => {{1,9},{2,4},{3,5},{6,7},{8,10}}
 => ?
 => ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => {{1,8},{2,4},{3,5},{6,7},{9,10}}
 => ?
 => ? = 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => {{1,7},{2,4},{3,5},{6,8},{9,10}}
 => ?
 => ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => {{1,6},{2,4},{3,5},{7,8},{9,10}}
 => ?
 => ? = 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => {{1,5},{2,4},{3,6},{7,8},{9,10}}
 => ?
 => ? = 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => {{1,4},{2,5},{3,6},{7,8},{9,10}}
 => ?
 => ? = 3
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => {{1,3},{2,5},{4,6},{7,8},{9,10}}
 => ?
 => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => {{1,2},{3,5},{4,6},{7,8},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => {{1,3},{2,6},{4,5},{7,8},{9,10}}
 => ?
 => ? = 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => {{1,4},{2,6},{3,5},{7,8},{9,10}}
 => ?
 => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => {{1,5},{2,6},{3,4},{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}}
 => ? = 0
[(1,7),(2,5),(3,4),(6,8),(9,10)]
 => {{1,7},{2,5},{3,4},{6,8},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,9),(2,5),(3,4),(6,7),(8,10)]
 => {{1,9},{2,5},{3,4},{6,7},{8,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,10),(2,6),(3,4),(5,7),(8,9)]
 => {{1,10},{2,6},{3,4},{5,7},{8,9}}
 => ?
 => ? = 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
 => {{1,9},{2,6},{3,4},{5,7},{8,10}}
 => ?
 => ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
 => {{1,8},{2,6},{3,4},{5,7},{9,10}}
 => ?
 => ? = 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
 => {{1,7},{2,6},{3,4},{5,8},{9,10}}
 => ?
 => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
 => {{1,6},{2,7},{3,4},{5,8},{9,10}}
 => ?
 => ? = 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
 => {{1,5},{2,7},{3,4},{6,8},{9,10}}
 => ?
 => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
 => {{1,4},{2,7},{3,5},{6,8},{9,10}}
 => ?
 => ? = 3
[(1,3),(2,7),(4,5),(6,8),(9,10)]
 => {{1,3},{2,7},{4,5},{6,8},{9,10}}
 => ?
 => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
 => {{1,2},{3,7},{4,5},{6,8},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,3),(2,8),(4,5),(6,7),(9,10)]
 => {{1,3},{2,8},{4,5},{6,7},{9,10}}
 => ?
 => ? = 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
 => {{1,4},{2,8},{3,5},{6,7},{9,10}}
 => ?
 => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
 => {{1,5},{2,8},{3,4},{6,7},{9,10}}
 => ?
 => ? = 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
 => {{1,6},{2,8},{3,4},{5,7},{9,10}}
 => ?
 => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
 => {{1,7},{2,8},{3,4},{5,6},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,9),(2,7),(3,4),(5,6),(8,10)]
 => {{1,9},{2,7},{3,4},{5,6},{8,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,10),(2,8),(3,4),(5,6),(7,9)]
 => {{1,10},{2,8},{3,4},{5,6},{7,9}}
 => ?
 => ? = 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
 => {{1,9},{2,8},{3,4},{5,6},{7,10}}
 => ?
 => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
 => {{1,8},{2,9},{3,4},{5,6},{7,10}}
 => ?
 => ? = 3
[(1,7),(2,9),(3,4),(5,6),(8,10)]
 => {{1,7},{2,9},{3,4},{5,6},{8,10}}
 => ?
 => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
 => {{1,6},{2,9},{3,4},{5,7},{8,10}}
 => ?
 => ? = 3
[(1,5),(2,9),(3,4),(6,7),(8,10)]
 => {{1,5},{2,9},{3,4},{6,7},{8,10}}
 => ?
 => ? = 2

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 5 compositions to match this statistic)

Mapping

Mp00092: Perfect matchings —to set partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 50%

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}}
 => 1
[(1,4),(2,3)]
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 0
[(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}}
 => 1
[(1,4),(2,3),(5,6)]
 => {{1,4},{2,3},{5,6}}
 => {{1,3},{2,4},{5,6}}
 => 0
[(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}}
 => 0
[(1,6),(2,4),(3,5)]
 => {{1,6},{2,4},{3,5}}
 => {{1,5},{2,4},{3,6}}
 => 1
[(1,5),(2,4),(3,6)]
 => {{1,5},{2,4},{3,6}}
 => {{1,6},{2,4},{3,5}}
 => 2
[(1,4),(2,5),(3,6)]
 => {{1,4},{2,5},{3,6}}
 => {{1,6},{2,5},{3,4}}
 => 3
[(1,3),(2,5),(4,6)]
 => {{1,3},{2,5},{4,6}}
 => {{1,6},{2,3},{4,5}}
 => 2
[(1,2),(3,5),(4,6)]
 => {{1,2},{3,5},{4,6}}
 => {{1,2},{3,6},{4,5}}
 => 1
[(1,2),(3,6),(4,5)]
 => {{1,2},{3,6},{4,5}}
 => {{1,2},{3,5},{4,6}}
 => 0
[(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}}
 => 2
[(1,5),(2,6),(3,4)]
 => {{1,5},{2,6},{3,4}}
 => {{1,4},{2,6},{3,5}}
 => 1
[(1,6),(2,5),(3,4)]
 => {{1,6},{2,5},{3,4}}
 => {{1,4},{2,5},{3,6}}
 => 0
[(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}}
 => 1
[(1,4),(2,3),(5,6),(7,8)]
 => {{1,4},{2,3},{5,6},{7,8}}
 => {{1,3},{2,4},{5,6},{7,8}}
 => 0
[(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}}
 => 0
[(1,7),(2,3),(4,5),(6,8)]
 => {{1,7},{2,3},{4,5},{6,8}}
 => {{1,3},{2,5},{4,8},{6,7}}
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => {{1,8},{2,3},{4,5},{6,7}}
 => {{1,3},{2,5},{4,7},{6,8}}
 => 0
[(1,8),(2,4),(3,5),(6,7)]
 => {{1,8},{2,4},{3,5},{6,7}}
 => {{1,5},{2,4},{3,7},{6,8}}
 => 1
[(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}}
 => 1
[(1,5),(2,4),(3,6),(7,8)]
 => {{1,5},{2,4},{3,6},{7,8}}
 => {{1,6},{2,4},{3,5},{7,8}}
 => 2
[(1,4),(2,5),(3,6),(7,8)]
 => {{1,4},{2,5},{3,6},{7,8}}
 => {{1,6},{2,5},{3,4},{7,8}}
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => {{1,3},{2,5},{4,6},{7,8}}
 => {{1,6},{2,3},{4,5},{7,8}}
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => {{1,2},{3,5},{4,6},{7,8}}
 => {{1,2},{3,6},{4,5},{7,8}}
 => 1
[(1,2),(3,6),(4,5),(7,8)]
 => {{1,2},{3,6},{4,5},{7,8}}
 => {{1,2},{3,5},{4,6},{7,8}}
 => 0
[(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}}
 => 2
[(1,5),(2,6),(3,4),(7,8)]
 => {{1,5},{2,6},{3,4},{7,8}}
 => {{1,4},{2,6},{3,5},{7,8}}
 => 1
[(1,6),(2,5),(3,4),(7,8)]
 => {{1,6},{2,5},{3,4},{7,8}}
 => {{1,4},{2,5},{3,6},{7,8}}
 => 0
[(1,7),(2,5),(3,4),(6,8)]
 => {{1,7},{2,5},{3,4},{6,8}}
 => {{1,4},{2,5},{3,8},{6,7}}
 => 1
[(1,8),(2,5),(3,4),(6,7)]
 => {{1,8},{2,5},{3,4},{6,7}}
 => {{1,4},{2,5},{3,7},{6,8}}
 => 0
[(1,8),(2,6),(3,4),(5,7)]
 => {{1,8},{2,6},{3,4},{5,7}}
 => {{1,4},{2,7},{3,6},{5,8}}
 => 1
[(1,7),(2,6),(3,4),(5,8)]
 => {{1,7},{2,6},{3,4},{5,8}}
 => {{1,4},{2,8},{3,6},{5,7}}
 => 2
[(1,6),(2,7),(3,4),(5,8)]
 => {{1,6},{2,7},{3,4},{5,8}}
 => {{1,4},{2,8},{3,7},{5,6}}
 => 3
[(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}}
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => {{1,3},{2,7},{4,5},{6,8}}
 => {{1,5},{2,3},{4,8},{6,7}}
 => 2
[(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}}
 => 0
[(1,3),(2,8),(4,5),(6,7)]
 => {{1,3},{2,8},{4,5},{6,7}}
 => {{1,5},{2,3},{4,7},{6,8}}
 => 1
[(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}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => {{1,5},{2,3},{4,6},{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}}
 => ? = 0
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => {{1,7},{2,3},{4,5},{6,8},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => {{1,9},{2,3},{4,5},{6,7},{8,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,10),(2,4),(3,5),(6,7),(8,9)]
 => {{1,10},{2,4},{3,5},{6,7},{8,9}}
 => ?
 => ? = 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
 => {{1,9},{2,4},{3,5},{6,7},{8,10}}
 => ?
 => ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => {{1,8},{2,4},{3,5},{6,7},{9,10}}
 => ?
 => ? = 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => {{1,7},{2,4},{3,5},{6,8},{9,10}}
 => ?
 => ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => {{1,6},{2,4},{3,5},{7,8},{9,10}}
 => ?
 => ? = 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => {{1,5},{2,4},{3,6},{7,8},{9,10}}
 => ?
 => ? = 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => {{1,4},{2,5},{3,6},{7,8},{9,10}}
 => ?
 => ? = 3
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => {{1,3},{2,5},{4,6},{7,8},{9,10}}
 => ?
 => ? = 2
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => {{1,2},{3,5},{4,6},{7,8},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => {{1,3},{2,6},{4,5},{7,8},{9,10}}
 => ?
 => ? = 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => {{1,4},{2,6},{3,5},{7,8},{9,10}}
 => ?
 => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => {{1,5},{2,6},{3,4},{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}}
 => ? = 0
[(1,7),(2,5),(3,4),(6,8),(9,10)]
 => {{1,7},{2,5},{3,4},{6,8},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,9),(2,5),(3,4),(6,7),(8,10)]
 => {{1,9},{2,5},{3,4},{6,7},{8,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,10),(2,6),(3,4),(5,7),(8,9)]
 => {{1,10},{2,6},{3,4},{5,7},{8,9}}
 => ?
 => ? = 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
 => {{1,9},{2,6},{3,4},{5,7},{8,10}}
 => ?
 => ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
 => {{1,8},{2,6},{3,4},{5,7},{9,10}}
 => ?
 => ? = 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
 => {{1,7},{2,6},{3,4},{5,8},{9,10}}
 => ?
 => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
 => {{1,6},{2,7},{3,4},{5,8},{9,10}}
 => ?
 => ? = 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
 => {{1,5},{2,7},{3,4},{6,8},{9,10}}
 => ?
 => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
 => {{1,4},{2,7},{3,5},{6,8},{9,10}}
 => ?
 => ? = 3
[(1,3),(2,7),(4,5),(6,8),(9,10)]
 => {{1,3},{2,7},{4,5},{6,8},{9,10}}
 => ?
 => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
 => {{1,2},{3,7},{4,5},{6,8},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,3),(2,8),(4,5),(6,7),(9,10)]
 => {{1,3},{2,8},{4,5},{6,7},{9,10}}
 => ?
 => ? = 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
 => {{1,4},{2,8},{3,5},{6,7},{9,10}}
 => ?
 => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
 => {{1,5},{2,8},{3,4},{6,7},{9,10}}
 => ?
 => ? = 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
 => {{1,6},{2,8},{3,4},{5,7},{9,10}}
 => ?
 => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
 => {{1,7},{2,8},{3,4},{5,6},{9,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,9),(2,7),(3,4),(5,6),(8,10)]
 => {{1,9},{2,7},{3,4},{5,6},{8,10}}
 => ?
 => ? = 1
[(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}}
 => ? = 0
[(1,10),(2,8),(3,4),(5,6),(7,9)]
 => {{1,10},{2,8},{3,4},{5,6},{7,9}}
 => ?
 => ? = 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
 => {{1,9},{2,8},{3,4},{5,6},{7,10}}
 => ?
 => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
 => {{1,8},{2,9},{3,4},{5,6},{7,10}}
 => ?
 => ? = 3
[(1,7),(2,9),(3,4),(5,6),(8,10)]
 => {{1,7},{2,9},{3,4},{5,6},{8,10}}
 => ?
 => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
 => {{1,6},{2,9},{3,4},{5,7},{8,10}}
 => ?
 => ? = 3
[(1,5),(2,9),(3,4),(6,7),(8,10)]
 => {{1,5},{2,9},{3,4},{6,7},{8,10}}
 => ?
 => ? = 2

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: St000123
(load all 7 compositions to match this statistic)

Mapping

Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000123: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 44%

Values

[(1,2)]
 => [(1,2)]
 => [2,1] => [2,1] => 0
[(1,2),(3,4)]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)]
 => [(1,4),(2,3)]
 => [4,3,2,1] => [3,2,4,1] => 1
[(1,4),(2,3)]
 => [(1,3),(2,4)]
 => [3,4,1,2] => [3,1,4,2] => 0
[(1,2),(3,4),(5,6)]
 => [(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)]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [3,2,4,1,6,5] => 1
[(1,4),(2,3),(5,6)]
 => [(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,1,4,2,6,5] => 0
[(1,5),(2,3),(4,6)]
 => [(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1
[(1,6),(2,3),(4,5)]
 => [(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,1,5,2,6,4] => 0
[(1,6),(2,4),(3,5)]
 => [(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [4,2,5,1,6,3] => 1
[(1,5),(2,4),(3,6)]
 => [(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [4,2,5,3,6,1] => 2
[(1,4),(2,5),(3,6)]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [4,3,5,2,6,1] => 3
[(1,3),(2,5),(4,6)]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [3,2,5,4,6,1] => 2
[(1,2),(3,5),(4,6)]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1
[(1,2),(3,6),(4,5)]
 => [(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,5,3,6,4] => 0
[(1,3),(2,6),(4,5)]
 => [(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1
[(1,4),(2,6),(3,5)]
 => [(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [4,3,5,1,6,2] => 2
[(1,5),(2,6),(3,4)]
 => [(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [4,1,5,3,6,2] => 1
[(1,6),(2,5),(3,4)]
 => [(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
 => [(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)]
 => [(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
[(1,4),(2,3),(5,6),(7,8)]
 => [(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
[(1,5),(2,3),(4,6),(7,8)]
 => [(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
[(1,6),(2,3),(4,5),(7,8)]
 => [(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
[(1,7),(2,3),(4,5),(6,8)]
 => [(1,3),(2,5),(4,8),(6,7)]
 => [3,5,1,8,2,7,6,4] => [3,1,5,2,7,6,8,4] => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [3,1,5,2,7,4,8,6] => 0
[(1,8),(2,4),(3,5),(6,7)]
 => [(1,5),(2,4),(3,7),(6,8)]
 => [5,4,7,2,1,8,3,6] => [4,2,5,1,7,3,8,6] => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [(1,5),(2,4),(3,8),(6,7)]
 => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [(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
[(1,5),(2,4),(3,6),(7,8)]
 => [(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
[(1,4),(2,5),(3,6),(7,8)]
 => [(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
[(1,3),(2,5),(4,6),(7,8)]
 => [(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
[(1,2),(3,5),(4,6),(7,8)]
 => [(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
[(1,2),(3,6),(4,5),(7,8)]
 => [(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
[(1,3),(2,6),(4,5),(7,8)]
 => [(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
[(1,4),(2,6),(3,5),(7,8)]
 => [(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
[(1,5),(2,6),(3,4),(7,8)]
 => [(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
[(1,6),(2,5),(3,4),(7,8)]
 => [(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
[(1,7),(2,5),(3,4),(6,8)]
 => [(1,4),(2,5),(3,8),(6,7)]
 => [4,5,8,1,2,7,6,3] => [4,1,5,2,7,6,8,3] => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [(1,4),(2,5),(3,7),(6,8)]
 => [4,5,7,1,2,8,3,6] => [4,1,5,2,7,3,8,6] => 0
[(1,8),(2,6),(3,4),(5,7)]
 => [(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [4,1,6,3,7,2,8,5] => 1
[(1,7),(2,6),(3,4),(5,8)]
 => [(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [4,1,6,3,7,5,8,2] => 2
[(1,6),(2,7),(3,4),(5,8)]
 => [(1,4),(2,8),(3,7),(5,6)]
 => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [(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
[(1,4),(2,7),(3,5),(6,8)]
 => [(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [(1,5),(2,3),(4,8),(6,7)]
 => [5,3,2,8,1,7,6,4] => [3,2,5,1,7,6,8,4] => 2
[(1,2),(3,7),(4,5),(6,8)]
 => [(1,2),(3,5),(4,8),(6,7)]
 => [2,1,5,8,3,7,6,4] => [2,1,5,3,7,6,8,4] => 1
[(1,2),(3,8),(4,5),(6,7)]
 => [(1,2),(3,5),(4,7),(6,8)]
 => [2,1,5,7,3,8,4,6] => [2,1,5,3,7,4,8,6] => 0
[(1,3),(2,8),(4,5),(6,7)]
 => [(1,5),(2,3),(4,7),(6,8)]
 => [5,3,2,7,1,8,4,6] => [3,2,5,1,7,4,8,6] => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [(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
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [(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
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [(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
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [3,6,1,5,4,2,8,7,10,9] => ? => ? = 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [(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
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [(1,3),(2,5),(4,8),(6,7),(9,10)]
 => [3,5,1,8,2,7,6,4,10,9] => ? => ? = 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [(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
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => [(1,3),(2,5),(4,7),(6,10),(8,9)]
 => [3,5,1,7,2,10,4,9,8,6] => ? => ? = 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [(1,3),(2,5),(4,7),(6,9),(8,10)]
 => [3,5,1,7,2,9,4,10,6,8] => [3,1,5,2,7,4,9,6,10,8] => ? = 0
[(1,10),(2,4),(3,5),(6,7),(8,9)]
 => [(1,5),(2,4),(3,7),(6,9),(8,10)]
 => [5,4,7,2,1,9,3,10,6,8] => ? => ? = 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
 => [(1,5),(2,4),(3,7),(6,10),(8,9)]
 => [5,4,7,2,1,10,3,9,8,6] => ? => ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [(1,5),(2,4),(3,7),(6,8),(9,10)]
 => [5,4,7,2,1,8,3,6,10,9] => ? => ? = 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [(1,5),(2,4),(3,8),(6,7),(9,10)]
 => [5,4,8,2,1,7,6,3,10,9] => ? => ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [5,4,6,2,1,3,8,7,10,9] => ? => ? = 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [6,4,5,2,3,1,8,7,10,9] => [4,2,5,3,6,1,8,7,10,9] => ? = 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [(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
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => [(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
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [(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
[(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [(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
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [5,3,2,6,1,4,8,7,10,9] => ? => ? = 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [(1,5),(2,6),(3,4),(7,8),(9,10)]
 => [5,6,4,3,1,2,8,7,10,9] => ? => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => [(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [4,6,5,1,3,2,8,7,10,9] => ? => ? = 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [(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
[(1,7),(2,5),(3,4),(6,8),(9,10)]
 => [(1,4),(2,5),(3,8),(6,7),(9,10)]
 => [4,5,8,1,2,7,6,3,10,9] => ? => ? = 1
[(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [(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
[(1,9),(2,5),(3,4),(6,7),(8,10)]
 => [(1,4),(2,5),(3,7),(6,10),(8,9)]
 => [4,5,7,1,2,10,3,9,8,6] => ? => ? = 1
[(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [(1,4),(2,5),(3,7),(6,9),(8,10)]
 => [4,5,7,1,2,9,3,10,6,8] => [4,1,5,2,7,3,9,6,10,8] => ? = 0
[(1,10),(2,6),(3,4),(5,7),(8,9)]
 => [(1,4),(2,7),(3,6),(5,9),(8,10)]
 => [4,7,6,1,9,3,2,10,5,8] => ? => ? = 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
 => [(1,4),(2,7),(3,6),(5,10),(8,9)]
 => [4,7,6,1,10,3,2,9,8,5] => ? => ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
 => [(1,4),(2,7),(3,6),(5,8),(9,10)]
 => [4,7,6,1,8,3,2,5,10,9] => ? => ? = 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
 => [(1,4),(2,8),(3,6),(5,7),(9,10)]
 => [4,8,6,1,7,3,5,2,10,9] => ? => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
 => [(1,4),(2,8),(3,7),(5,6),(9,10)]
 => [4,8,7,1,6,5,3,2,10,9] => ? => ? = 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
 => [(1,4),(2,8),(3,5),(6,7),(9,10)]
 => [4,8,5,1,3,7,6,2,10,9] => ? => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
 => [(1,5),(2,8),(3,4),(6,7),(9,10)]
 => [5,8,4,3,1,7,6,2,10,9] => ? => ? = 3
[(1,3),(2,7),(4,5),(6,8),(9,10)]
 => [(1,5),(2,3),(4,8),(6,7),(9,10)]
 => [5,3,2,8,1,7,6,4,10,9] => ? => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
 => [(1,2),(3,5),(4,8),(6,7),(9,10)]
 => [2,1,5,8,3,7,6,4,10,9] => ? => ? = 1
[(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [(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),(2,8),(4,5),(6,7),(9,10)]
 => [(1,5),(2,3),(4,7),(6,8),(9,10)]
 => [5,3,2,7,1,8,4,6,10,9] => ? => ? = 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
 => [(1,5),(2,7),(3,4),(6,8),(9,10)]
 => [5,7,4,3,1,8,2,6,10,9] => ? => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
 => [(1,4),(2,7),(3,5),(6,8),(9,10)]
 => [4,7,5,1,3,8,2,6,10,9] => ? => ? = 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
 => [(1,4),(2,7),(3,8),(5,6),(9,10)]
 => [4,7,8,1,6,5,2,3,10,9] => ? => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
 => [(1,4),(2,6),(3,8),(5,7),(9,10)]
 => [4,6,8,1,7,2,5,3,10,9] => ? => ? = 1
[(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [(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
[(1,9),(2,7),(3,4),(5,6),(8,10)]
 => [(1,4),(2,6),(3,7),(5,10),(8,9)]
 => [4,6,7,1,10,2,3,9,8,5] => ? => ? = 1
[(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [(1,4),(2,6),(3,7),(5,9),(8,10)]
 => [4,6,7,1,9,2,3,10,5,8] => [4,1,6,2,7,3,9,5,10,8] => ? = 0
[(1,10),(2,8),(3,4),(5,6),(7,9)]
 => [(1,4),(2,6),(3,9),(5,8),(7,10)]
 => [4,6,9,1,8,2,10,5,3,7] => ? => ? = 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
 => [(1,4),(2,6),(3,10),(5,8),(7,9)]
 => [4,6,10,1,8,2,9,5,7,3] => ? => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
 => [(1,4),(2,6),(3,10),(5,9),(7,8)]
 => [4,6,10,1,9,2,8,7,5,3] => ? => ? = 3
[(1,7),(2,9),(3,4),(5,6),(8,10)]
 => [(1,4),(2,6),(3,10),(5,7),(8,9)]
 => [4,6,10,1,7,2,5,9,8,3] => ? => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
 => [(1,4),(2,7),(3,10),(5,6),(8,9)]
 => [4,7,10,1,6,5,2,9,8,3] => ? => ? = 3
[(1,5),(2,9),(3,4),(6,7),(8,10)]
 => [(1,4),(2,7),(3,5),(6,10),(8,9)]
 => [4,7,5,1,3,10,2,9,8,6] => ? => ? = 2

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

Mapping

Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 44%

Values

[(1,2)]
 => [(1,2)]
 => [2,1] => [2,1] => 0
[(1,2),(3,4)]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)]
 => [(1,4),(2,3)]
 => [4,3,2,1] => [3,2,4,1] => 1
[(1,4),(2,3)]
 => [(1,3),(2,4)]
 => [3,4,1,2] => [3,1,4,2] => 0
[(1,2),(3,4),(5,6)]
 => [(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)]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [3,2,4,1,6,5] => 1
[(1,4),(2,3),(5,6)]
 => [(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,1,4,2,6,5] => 0
[(1,5),(2,3),(4,6)]
 => [(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [3,1,5,4,6,2] => 1
[(1,6),(2,3),(4,5)]
 => [(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,1,5,2,6,4] => 0
[(1,6),(2,4),(3,5)]
 => [(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [4,2,5,1,6,3] => 1
[(1,5),(2,4),(3,6)]
 => [(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [4,2,5,3,6,1] => 2
[(1,4),(2,5),(3,6)]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [4,3,5,2,6,1] => 3
[(1,3),(2,5),(4,6)]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [3,2,5,4,6,1] => 2
[(1,2),(3,5),(4,6)]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1
[(1,2),(3,6),(4,5)]
 => [(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,5,3,6,4] => 0
[(1,3),(2,6),(4,5)]
 => [(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1
[(1,4),(2,6),(3,5)]
 => [(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [4,3,5,1,6,2] => 2
[(1,5),(2,6),(3,4)]
 => [(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [4,1,5,3,6,2] => 1
[(1,6),(2,5),(3,4)]
 => [(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
 => [(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)]
 => [(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
[(1,4),(2,3),(5,6),(7,8)]
 => [(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
[(1,5),(2,3),(4,6),(7,8)]
 => [(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
[(1,6),(2,3),(4,5),(7,8)]
 => [(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
[(1,7),(2,3),(4,5),(6,8)]
 => [(1,3),(2,5),(4,8),(6,7)]
 => [3,5,1,8,2,7,6,4] => [3,1,5,2,7,6,8,4] => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [(1,3),(2,5),(4,7),(6,8)]
 => [3,5,1,7,2,8,4,6] => [3,1,5,2,7,4,8,6] => 0
[(1,8),(2,4),(3,5),(6,7)]
 => [(1,5),(2,4),(3,7),(6,8)]
 => [5,4,7,2,1,8,3,6] => [4,2,5,1,7,3,8,6] => 1
[(1,7),(2,4),(3,5),(6,8)]
 => [(1,5),(2,4),(3,8),(6,7)]
 => [5,4,8,2,1,7,6,3] => [4,2,5,1,7,6,8,3] => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [(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
[(1,5),(2,4),(3,6),(7,8)]
 => [(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
[(1,4),(2,5),(3,6),(7,8)]
 => [(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
[(1,3),(2,5),(4,6),(7,8)]
 => [(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
[(1,2),(3,5),(4,6),(7,8)]
 => [(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
[(1,2),(3,6),(4,5),(7,8)]
 => [(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
[(1,3),(2,6),(4,5),(7,8)]
 => [(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
[(1,4),(2,6),(3,5),(7,8)]
 => [(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
[(1,5),(2,6),(3,4),(7,8)]
 => [(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
[(1,6),(2,5),(3,4),(7,8)]
 => [(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
[(1,7),(2,5),(3,4),(6,8)]
 => [(1,4),(2,5),(3,8),(6,7)]
 => [4,5,8,1,2,7,6,3] => [4,1,5,2,7,6,8,3] => 1
[(1,8),(2,5),(3,4),(6,7)]
 => [(1,4),(2,5),(3,7),(6,8)]
 => [4,5,7,1,2,8,3,6] => [4,1,5,2,7,3,8,6] => 0
[(1,8),(2,6),(3,4),(5,7)]
 => [(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [4,1,6,3,7,2,8,5] => 1
[(1,7),(2,6),(3,4),(5,8)]
 => [(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [4,1,6,3,7,5,8,2] => 2
[(1,6),(2,7),(3,4),(5,8)]
 => [(1,4),(2,8),(3,7),(5,6)]
 => [4,8,7,1,6,5,3,2] => [4,1,6,5,7,3,8,2] => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [(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
[(1,4),(2,7),(3,5),(6,8)]
 => [(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [(1,5),(2,3),(4,8),(6,7)]
 => [5,3,2,8,1,7,6,4] => [3,2,5,1,7,6,8,4] => 2
[(1,2),(3,7),(4,5),(6,8)]
 => [(1,2),(3,5),(4,8),(6,7)]
 => [2,1,5,8,3,7,6,4] => [2,1,5,3,7,6,8,4] => 1
[(1,2),(3,8),(4,5),(6,7)]
 => [(1,2),(3,5),(4,7),(6,8)]
 => [2,1,5,7,3,8,4,6] => [2,1,5,3,7,4,8,6] => 0
[(1,3),(2,8),(4,5),(6,7)]
 => [(1,5),(2,3),(4,7),(6,8)]
 => [5,3,2,7,1,8,4,6] => [3,2,5,1,7,4,8,6] => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [(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
[(1,3),(2,4),(5,6),(7,8),(9,10)]
 => [(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
[(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [(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
[(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [3,6,1,5,4,2,8,7,10,9] => ? => ? = 1
[(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [(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
[(1,7),(2,3),(4,5),(6,8),(9,10)]
 => [(1,3),(2,5),(4,8),(6,7),(9,10)]
 => [3,5,1,8,2,7,6,4,10,9] => ? => ? = 1
[(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [(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
[(1,9),(2,3),(4,5),(6,7),(8,10)]
 => [(1,3),(2,5),(4,7),(6,10),(8,9)]
 => [3,5,1,7,2,10,4,9,8,6] => ? => ? = 1
[(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [(1,3),(2,5),(4,7),(6,9),(8,10)]
 => [3,5,1,7,2,9,4,10,6,8] => [3,1,5,2,7,4,9,6,10,8] => ? = 0
[(1,10),(2,4),(3,5),(6,7),(8,9)]
 => [(1,5),(2,4),(3,7),(6,9),(8,10)]
 => [5,4,7,2,1,9,3,10,6,8] => ? => ? = 1
[(1,9),(2,4),(3,5),(6,7),(8,10)]
 => [(1,5),(2,4),(3,7),(6,10),(8,9)]
 => [5,4,7,2,1,10,3,9,8,6] => ? => ? = 2
[(1,8),(2,4),(3,5),(6,7),(9,10)]
 => [(1,5),(2,4),(3,7),(6,8),(9,10)]
 => [5,4,7,2,1,8,3,6,10,9] => ? => ? = 1
[(1,7),(2,4),(3,5),(6,8),(9,10)]
 => [(1,5),(2,4),(3,8),(6,7),(9,10)]
 => [5,4,8,2,1,7,6,3,10,9] => ? => ? = 2
[(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [5,4,6,2,1,3,8,7,10,9] => ? => ? = 1
[(1,5),(2,4),(3,6),(7,8),(9,10)]
 => [(1,6),(2,4),(3,5),(7,8),(9,10)]
 => [6,4,5,2,3,1,8,7,10,9] => [4,2,5,3,6,1,8,7,10,9] => ? = 2
[(1,4),(2,5),(3,6),(7,8),(9,10)]
 => [(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
[(1,3),(2,5),(4,6),(7,8),(9,10)]
 => [(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
[(1,2),(3,5),(4,6),(7,8),(9,10)]
 => [(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
[(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [(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
[(1,3),(2,6),(4,5),(7,8),(9,10)]
 => [(1,5),(2,3),(4,6),(7,8),(9,10)]
 => [5,3,2,6,1,4,8,7,10,9] => ? => ? = 1
[(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [(1,5),(2,6),(3,4),(7,8),(9,10)]
 => [5,6,4,3,1,2,8,7,10,9] => ? => ? = 2
[(1,5),(2,6),(3,4),(7,8),(9,10)]
 => [(1,4),(2,6),(3,5),(7,8),(9,10)]
 => [4,6,5,1,3,2,8,7,10,9] => ? => ? = 1
[(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [(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
[(1,7),(2,5),(3,4),(6,8),(9,10)]
 => [(1,4),(2,5),(3,8),(6,7),(9,10)]
 => [4,5,8,1,2,7,6,3,10,9] => ? => ? = 1
[(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [(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
[(1,9),(2,5),(3,4),(6,7),(8,10)]
 => [(1,4),(2,5),(3,7),(6,10),(8,9)]
 => [4,5,7,1,2,10,3,9,8,6] => ? => ? = 1
[(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [(1,4),(2,5),(3,7),(6,9),(8,10)]
 => [4,5,7,1,2,9,3,10,6,8] => [4,1,5,2,7,3,9,6,10,8] => ? = 0
[(1,10),(2,6),(3,4),(5,7),(8,9)]
 => [(1,4),(2,7),(3,6),(5,9),(8,10)]
 => [4,7,6,1,9,3,2,10,5,8] => ? => ? = 1
[(1,9),(2,6),(3,4),(5,7),(8,10)]
 => [(1,4),(2,7),(3,6),(5,10),(8,9)]
 => [4,7,6,1,10,3,2,9,8,5] => ? => ? = 2
[(1,8),(2,6),(3,4),(5,7),(9,10)]
 => [(1,4),(2,7),(3,6),(5,8),(9,10)]
 => [4,7,6,1,8,3,2,5,10,9] => ? => ? = 1
[(1,7),(2,6),(3,4),(5,8),(9,10)]
 => [(1,4),(2,8),(3,6),(5,7),(9,10)]
 => [4,8,6,1,7,3,5,2,10,9] => ? => ? = 2
[(1,6),(2,7),(3,4),(5,8),(9,10)]
 => [(1,4),(2,8),(3,7),(5,6),(9,10)]
 => [4,8,7,1,6,5,3,2,10,9] => ? => ? = 3
[(1,5),(2,7),(3,4),(6,8),(9,10)]
 => [(1,4),(2,8),(3,5),(6,7),(9,10)]
 => [4,8,5,1,3,7,6,2,10,9] => ? => ? = 2
[(1,4),(2,7),(3,5),(6,8),(9,10)]
 => [(1,5),(2,8),(3,4),(6,7),(9,10)]
 => [5,8,4,3,1,7,6,2,10,9] => ? => ? = 3
[(1,3),(2,7),(4,5),(6,8),(9,10)]
 => [(1,5),(2,3),(4,8),(6,7),(9,10)]
 => [5,3,2,8,1,7,6,4,10,9] => ? => ? = 2
[(1,2),(3,7),(4,5),(6,8),(9,10)]
 => [(1,2),(3,5),(4,8),(6,7),(9,10)]
 => [2,1,5,8,3,7,6,4,10,9] => ? => ? = 1
[(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [(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),(2,8),(4,5),(6,7),(9,10)]
 => [(1,5),(2,3),(4,7),(6,8),(9,10)]
 => [5,3,2,7,1,8,4,6,10,9] => ? => ? = 1
[(1,4),(2,8),(3,5),(6,7),(9,10)]
 => [(1,5),(2,7),(3,4),(6,8),(9,10)]
 => [5,7,4,3,1,8,2,6,10,9] => ? => ? = 2
[(1,5),(2,8),(3,4),(6,7),(9,10)]
 => [(1,4),(2,7),(3,5),(6,8),(9,10)]
 => [4,7,5,1,3,8,2,6,10,9] => ? => ? = 1
[(1,6),(2,8),(3,4),(5,7),(9,10)]
 => [(1,4),(2,7),(3,8),(5,6),(9,10)]
 => [4,7,8,1,6,5,2,3,10,9] => ? => ? = 2
[(1,7),(2,8),(3,4),(5,6),(9,10)]
 => [(1,4),(2,6),(3,8),(5,7),(9,10)]
 => [4,6,8,1,7,2,5,3,10,9] => ? => ? = 1
[(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [(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
[(1,9),(2,7),(3,4),(5,6),(8,10)]
 => [(1,4),(2,6),(3,7),(5,10),(8,9)]
 => [4,6,7,1,10,2,3,9,8,5] => ? => ? = 1
[(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [(1,4),(2,6),(3,7),(5,9),(8,10)]
 => [4,6,7,1,9,2,3,10,5,8] => [4,1,6,2,7,3,9,5,10,8] => ? = 0
[(1,10),(2,8),(3,4),(5,6),(7,9)]
 => [(1,4),(2,6),(3,9),(5,8),(7,10)]
 => [4,6,9,1,8,2,10,5,3,7] => ? => ? = 1
[(1,9),(2,8),(3,4),(5,6),(7,10)]
 => [(1,4),(2,6),(3,10),(5,8),(7,9)]
 => [4,6,10,1,8,2,9,5,7,3] => ? => ? = 2
[(1,8),(2,9),(3,4),(5,6),(7,10)]
 => [(1,4),(2,6),(3,10),(5,9),(7,8)]
 => [4,6,10,1,9,2,8,7,5,3] => ? => ? = 3
[(1,7),(2,9),(3,4),(5,6),(8,10)]
 => [(1,4),(2,6),(3,10),(5,7),(8,9)]
 => [4,6,10,1,7,2,5,9,8,3] => ? => ? = 2
[(1,6),(2,9),(3,4),(5,7),(8,10)]
 => [(1,4),(2,7),(3,10),(5,6),(8,9)]
 => [4,7,10,1,6,5,2,9,8,3] => ? => ? = 3
[(1,5),(2,9),(3,4),(6,7),(8,10)]
 => [(1,4),(2,7),(3,5),(6,10),(8,9)]
 => [4,7,5,1,3,10,2,9,8,6] => ? => ? = 2

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 7 compositions to match this statistic)

Mapping

Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 50%

Values

[(1,2)]
 => [(1,2)]
 => [2,1] => [2,1] => 0
[(1,2),(3,4)]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 0
[(1,3),(2,4)]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [4,3,1,2] => 1
[(1,4),(2,3)]
 => [(1,3),(2,4)]
 => [3,4,1,2] => [3,4,1,2] => 0
[(1,2),(3,4),(5,6)]
 => [(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)]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,3,1,2,6,5] => 1
[(1,4),(2,3),(5,6)]
 => [(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,4,1,2,6,5] => 0
[(1,5),(2,3),(4,6)]
 => [(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [3,6,1,5,2,4] => 1
[(1,6),(2,3),(4,5)]
 => [(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,5,1,6,2,4] => 0
[(1,6),(2,4),(3,5)]
 => [(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [5,4,6,1,2,3] => 1
[(1,5),(2,4),(3,6)]
 => [(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [6,4,5,1,2,3] => 2
[(1,4),(2,5),(3,6)]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 3
[(1,3),(2,5),(4,6)]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,3,1,5,2,4] => 2
[(1,2),(3,5),(4,6)]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,3,4] => 1
[(1,2),(3,6),(4,5)]
 => [(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,5,6,3,4] => 0
[(1,3),(2,6),(4,5)]
 => [(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [5,3,1,6,2,4] => 1
[(1,4),(2,6),(3,5)]
 => [(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [5,6,4,1,2,3] => 2
[(1,5),(2,6),(3,4)]
 => [(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [4,6,5,1,2,3] => 1
[(1,6),(2,5),(3,4)]
 => [(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,5,6,1,2,3] => 0
[(1,2),(3,4),(5,6),(7,8)]
 => [(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)]
 => [(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
[(1,4),(2,3),(5,6),(7,8)]
 => [(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
[(1,5),(2,3),(4,6),(7,8)]
 => [(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
[(1,6),(2,3),(4,5),(7,8)]
 => [(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
[(1,7),(2,3),(4,5),(6,8)]
 => [(1,3),(2,5),(4,8),(6,7)]
 => [3,5,1,7,2,8,6,4] => [3,5,1,8,2,7,4,6] => ? = 1
[(1,8),(2,3),(4,5),(6,7)]
 => [(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
[(1,8),(2,4),(3,5),(6,7)]
 => [(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
[(1,7),(2,4),(3,5),(6,8)]
 => [(1,5),(2,4),(3,8),(6,7)]
 => [4,5,7,2,1,8,6,3] => [5,4,8,1,2,7,3,6] => ? = 2
[(1,6),(2,4),(3,5),(7,8)]
 => [(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
[(1,5),(2,4),(3,6),(7,8)]
 => [(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
[(1,4),(2,5),(3,6),(7,8)]
 => [(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
[(1,3),(2,5),(4,6),(7,8)]
 => [(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
[(1,2),(3,5),(4,6),(7,8)]
 => [(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
[(1,2),(3,6),(4,5),(7,8)]
 => [(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
[(1,3),(2,6),(4,5),(7,8)]
 => [(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
[(1,4),(2,6),(3,5),(7,8)]
 => [(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
[(1,5),(2,6),(3,4),(7,8)]
 => [(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
[(1,6),(2,5),(3,4),(7,8)]
 => [(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
[(1,7),(2,5),(3,4),(6,8)]
 => [(1,4),(2,5),(3,8),(6,7)]
 => [4,5,7,1,2,8,6,3] => [4,5,8,1,2,7,3,6] => ? = 1
[(1,8),(2,5),(3,4),(6,7)]
 => [(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
[(1,8),(2,6),(3,4),(5,7)]
 => [(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
[(1,7),(2,6),(3,4),(5,8)]
 => [(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
[(1,6),(2,7),(3,4),(5,8)]
 => [(1,4),(2,8),(3,7),(5,6)]
 => [4,6,7,1,8,5,3,2] => [4,8,7,1,6,2,3,5] => ? = 3
[(1,5),(2,7),(3,4),(6,8)]
 => [(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
[(1,4),(2,7),(3,5),(6,8)]
 => [(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
[(1,3),(2,7),(4,5),(6,8)]
 => [(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
[(1,2),(3,7),(4,5),(6,8)]
 => [(1,2),(3,5),(4,8),(6,7)]
 => [2,1,5,7,3,8,6,4] => [2,1,5,8,3,7,4,6] => ? = 1
[(1,2),(3,8),(4,5),(6,7)]
 => [(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),(2,8),(4,5),(6,7)]
 => [(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),(2,8),(3,5),(6,7)]
 => [(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
[(1,5),(2,8),(3,4),(6,7)]
 => [(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
[(1,6),(2,8),(3,4),(5,7)]
 => [(1,4),(2,7),(3,8),(5,6)]
 => [4,6,7,1,8,5,2,3] => [4,7,8,1,6,2,3,5] => ? = 2
[(1,7),(2,8),(3,4),(5,6)]
 => [(1,4),(2,6),(3,8),(5,7)]
 => [4,6,7,1,8,2,5,3] => [4,6,8,1,7,2,3,5] => ? = 1
[(1,8),(2,7),(3,4),(5,6)]
 => [(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
[(1,8),(2,7),(3,5),(4,6)]
 => [(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
[(1,7),(2,8),(3,5),(4,6)]
 => [(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
[(1,6),(2,8),(3,5),(4,7)]
 => [(1,7),(2,5),(3,8),(4,6)]
 => [5,6,7,8,2,4,1,3] => [7,5,8,6,1,2,3,4] => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
 => [(1,7),(2,6),(3,8),(4,5)]
 => [5,6,7,8,4,2,1,3] => [7,6,8,5,1,2,3,4] => ? = 4
[(1,4),(2,8),(3,6),(5,7)]
 => [(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
[(1,3),(2,8),(4,6),(5,7)]
 => [(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
[(1,2),(3,8),(4,6),(5,7)]
 => [(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
[(1,2),(3,7),(4,6),(5,8)]
 => [(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
[(1,3),(2,7),(4,6),(5,8)]
 => [(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
[(1,4),(2,7),(3,6),(5,8)]
 => [(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
[(1,5),(2,7),(3,6),(4,8)]
 => [(1,8),(2,6),(3,7),(4,5)]
 => [5,6,7,8,4,2,3,1] => [8,6,7,5,1,2,3,4] => ? = 5
[(1,6),(2,7),(3,5),(4,8)]
 => [(1,8),(2,5),(3,7),(4,6)]
 => [5,6,7,8,2,4,3,1] => [8,5,7,6,1,2,3,4] => ? = 4
[(1,7),(2,6),(3,5),(4,8)]
 => [(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
[(1,8),(2,6),(3,5),(4,7)]
 => [(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
[(1,8),(2,5),(3,6),(4,7)]
 => [(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
[(1,7),(2,5),(3,6),(4,8)]
 => [(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
[(1,6),(2,5),(3,7),(4,8)]
 => [(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
[(1,5),(2,6),(3,7),(4,8)]
 => [(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
[(1,4),(2,6),(3,7),(5,8)]
 => [(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
[(1,3),(2,6),(4,7),(5,8)]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [8,3,1,7,6,2,4,5] => ? = 4
[(1,2),(3,6),(4,7),(5,8)]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => [2,1,8,7,6,3,4,5] => ? = 3
[(1,2),(3,5),(4,7),(6,8)]
 => [(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),(2,5),(4,7),(6,8)]
 => [(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
[(1,4),(2,5),(3,7),(6,8)]
 => [(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
[(1,5),(2,4),(3,7),(6,8)]
 => [(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
[(1,6),(2,4),(3,7),(5,8)]
 => [(1,8),(2,4),(3,7),(5,6)]
 => [4,6,7,2,8,5,3,1] => [8,4,7,1,6,2,3,5] => ? = 4
[(1,7),(2,4),(3,6),(5,8)]
 => [(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),(2,4),(3,6),(5,7)]
 => [(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),(2,3),(4,6),(5,7)]
 => [(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
[(1,7),(2,3),(4,6),(5,8)]
 => [(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
[(1,6),(2,3),(4,7),(5,8)]
 => [(1,3),(2,8),(4,7),(5,6)]
 => [3,6,1,7,8,5,4,2] => [3,8,1,7,6,2,4,5] => ? = 3
[(1,3),(2,4),(5,7),(6,8)]
 => [(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
[(1,2),(3,4),(5,8),(6,7)]
 => [(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)]
 => [(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
[(1,8),(2,3),(4,7),(5,6)]
 => [(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
[(1,7),(2,4),(3,8),(5,6)]
 => [(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
[(1,4),(2,6),(3,8),(5,7)]
 => [(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
[(1,6),(2,5),(3,8),(4,7)]
 => [(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
[(1,2),(3,8),(4,7),(5,6)]
 => [(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
[(1,8),(2,7),(3,6),(4,5)]
 => [(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
[(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [(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
[(1,9),(2,8),(3,7),(4,6),(5,10)]
 => [(1,10),(2,6),(3,7),(4,8),(5,9)]
 => [6,7,8,9,10,2,3,4,5,1] => [10,6,7,8,9,1,2,3,4,5] => 4
[(1,6),(2,7),(3,8),(4,9),(5,10)]
 => [(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
[(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [(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
[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
 => [(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
[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [(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

Description

The number of nestings in the permutation.

Matching statistic: St000356

Mapping

Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●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] => [3,1,4,2] => [2,4,1,3] => 1
[(1,4),(2,3)]
 => [4,3,2,1] => [3,2,4,1] => [2,3,1,4] => 0
[(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] => [3,1,4,2,6,5] => [4,6,3,5,1,2] => 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [3,2,4,1,6,5] => [4,5,3,6,1,2] => 0
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [3,2,5,1,6,4] => [4,5,2,6,1,3] => 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [3,2,5,4,6,1] => [4,5,2,3,1,6] => 0
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [4,2,5,3,6,1] => [3,5,2,4,1,6] => 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [4,2,5,1,6,3] => [3,5,2,6,1,4] => 2
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [3,6,2,5,1,4] => 3
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => 2
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,5,3,6,4] => [5,6,2,4,1,3] => 1
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,5,4,6,3] => [5,6,2,3,1,4] => 0
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [3,1,5,4,6,2] => [4,6,2,3,1,5] => 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [4,1,5,3,6,2] => [3,6,2,4,1,5] => 2
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [4,3,5,1,6,2] => [3,4,2,6,1,5] => 1
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [4,3,5,2,6,1] => [3,4,2,5,1,6] => 0
[(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] => [3,1,4,2,6,5,8,7] => [6,8,5,7,3,4,1,2] => ? = 1
[(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => [3,2,4,1,6,5,8,7] => [6,7,5,8,3,4,1,2] => ? = 0
[(1,5),(2,3),(4,6),(7,8)]
 => [5,3,2,6,1,4,8,7] => [3,2,5,1,6,4,8,7] => [6,7,4,8,3,5,1,2] => ? = 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] => [6,7,4,5,3,8,1,2] => ? = 0
[(1,7),(2,3),(4,5),(6,8)]
 => [7,3,2,5,4,8,1,6] => [3,2,5,4,7,1,8,6] => [6,7,4,5,2,8,1,3] => ? = 1
[(1,8),(2,3),(4,5),(6,7)]
 => [8,3,2,5,4,7,6,1] => [3,2,5,4,7,6,8,1] => [6,7,4,5,2,3,1,8] => ? = 0
[(1,8),(2,4),(3,5),(6,7)]
 => [8,4,5,2,3,7,6,1] => [4,2,5,3,7,6,8,1] => [5,7,4,6,2,3,1,8] => ? = 1
[(1,7),(2,4),(3,5),(6,8)]
 => [7,4,5,2,3,8,1,6] => [4,2,5,3,7,1,8,6] => [5,7,4,6,2,8,1,3] => ? = 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] => [5,7,4,6,3,8,1,2] => ? = 1
[(1,5),(2,4),(3,6),(7,8)]
 => [5,4,6,2,1,3,8,7] => [4,2,5,1,6,3,8,7] => [5,7,4,8,3,6,1,2] => ? = 2
[(1,4),(2,5),(3,6),(7,8)]
 => [4,5,6,1,2,3,8,7] => [4,1,5,2,6,3,8,7] => [5,8,4,7,3,6,1,2] => ? = 3
[(1,3),(2,5),(4,6),(7,8)]
 => [3,5,1,6,2,4,8,7] => [3,1,5,2,6,4,8,7] => [6,8,4,7,3,5,1,2] => ? = 2
[(1,2),(3,5),(4,6),(7,8)]
 => [2,1,5,6,3,4,8,7] => [2,1,5,3,6,4,8,7] => [7,8,4,6,3,5,1,2] => ? = 1
[(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => [2,1,5,4,6,3,8,7] => [7,8,4,5,3,6,1,2] => ? = 0
[(1,3),(2,6),(4,5),(7,8)]
 => [3,6,1,5,4,2,8,7] => [3,1,5,4,6,2,8,7] => [6,8,4,5,3,7,1,2] => ? = 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] => [5,8,4,6,3,7,1,2] => ? = 2
[(1,5),(2,6),(3,4),(7,8)]
 => [5,6,4,3,1,2,8,7] => [4,3,5,1,6,2,8,7] => [5,6,4,8,3,7,1,2] => ? = 1
[(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => [4,3,5,2,6,1,8,7] => [5,6,4,7,3,8,1,2] => ? = 0
[(1,7),(2,5),(3,4),(6,8)]
 => [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => [5,6,4,7,2,8,1,3] => ? = 1
[(1,8),(2,5),(3,4),(6,7)]
 => [8,5,4,3,2,7,6,1] => [4,3,5,2,7,6,8,1] => [5,6,4,7,2,3,1,8] => ? = 0
[(1,8),(2,6),(3,4),(5,7)]
 => [8,6,4,3,7,2,5,1] => [4,3,6,2,7,5,8,1] => [5,6,3,7,2,4,1,8] => ? = 1
[(1,7),(2,6),(3,4),(5,8)]
 => [7,6,4,3,8,2,1,5] => [4,3,6,2,7,1,8,5] => [5,6,3,7,2,8,1,4] => ? = 2
[(1,6),(2,7),(3,4),(5,8)]
 => [6,7,4,3,8,1,2,5] => [4,3,6,1,7,2,8,5] => [5,6,3,8,2,7,1,4] => ? = 3
[(1,5),(2,7),(3,4),(6,8)]
 => [5,7,4,3,1,8,2,6] => [4,3,5,1,7,2,8,6] => [5,6,4,8,2,7,1,3] => ? = 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] => [5,8,4,6,2,7,1,3] => ? = 3
[(1,3),(2,7),(4,5),(6,8)]
 => [3,7,1,5,4,8,2,6] => [3,1,5,4,7,2,8,6] => [6,8,4,5,2,7,1,3] => ? = 2
[(1,2),(3,7),(4,5),(6,8)]
 => [2,1,7,5,4,8,3,6] => [2,1,5,4,7,3,8,6] => [7,8,4,5,2,6,1,3] => ? = 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] => [7,8,4,5,2,3,1,6] => ? = 0
[(1,3),(2,8),(4,5),(6,7)]
 => [3,8,1,5,4,7,6,2] => [3,1,5,4,7,6,8,2] => [6,8,4,5,2,3,1,7] => ? = 1
[(1,4),(2,8),(3,5),(6,7)]
 => [4,8,5,1,3,7,6,2] => [4,1,5,3,7,6,8,2] => [5,8,4,6,2,3,1,7] => ? = 2
[(1,5),(2,8),(3,4),(6,7)]
 => [5,8,4,3,1,7,6,2] => [4,3,5,1,7,6,8,2] => [5,6,4,8,2,3,1,7] => ? = 1
[(1,6),(2,8),(3,4),(5,7)]
 => [6,8,4,3,7,1,5,2] => [4,3,6,1,7,5,8,2] => [5,6,3,8,2,4,1,7] => ? = 2
[(1,7),(2,8),(3,4),(5,6)]
 => [7,8,4,3,6,5,1,2] => [4,3,6,5,7,1,8,2] => [5,6,3,4,2,8,1,7] => ? = 1
[(1,8),(2,7),(3,4),(5,6)]
 => [8,7,4,3,6,5,2,1] => [4,3,6,5,7,2,8,1] => [5,6,3,4,2,7,1,8] => ? = 0
[(1,8),(2,7),(3,5),(4,6)]
 => [8,7,5,6,3,4,2,1] => [5,3,6,4,7,2,8,1] => [4,6,3,5,2,7,1,8] => ? = 1
[(1,7),(2,8),(3,5),(4,6)]
 => [7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => [4,6,3,5,2,8,1,7] => ? = 2
[(1,6),(2,8),(3,5),(4,7)]
 => [6,8,5,7,3,1,4,2] => [5,3,6,1,7,4,8,2] => [4,6,3,8,2,5,1,7] => ? = 3
[(1,5),(2,8),(3,6),(4,7)]
 => [5,8,6,7,1,3,4,2] => [5,1,6,3,7,4,8,2] => [4,8,3,6,2,5,1,7] => ? = 4
[(1,4),(2,8),(3,6),(5,7)]
 => [4,8,6,1,7,3,5,2] => [4,1,6,3,7,5,8,2] => [5,8,3,6,2,4,1,7] => ? = 3
[(1,3),(2,8),(4,6),(5,7)]
 => [3,8,1,6,7,4,5,2] => [3,1,6,4,7,5,8,2] => [6,8,3,5,2,4,1,7] => ? = 2
[(1,2),(3,8),(4,6),(5,7)]
 => [2,1,8,6,7,4,5,3] => [2,1,6,4,7,5,8,3] => [7,8,3,5,2,4,1,6] => ? = 1
[(1,2),(3,7),(4,6),(5,8)]
 => [2,1,7,6,8,4,3,5] => [2,1,6,4,7,3,8,5] => [7,8,3,5,2,6,1,4] => ? = 2
[(1,3),(2,7),(4,6),(5,8)]
 => [3,7,1,6,8,4,2,5] => [3,1,6,4,7,2,8,5] => [6,8,3,5,2,7,1,4] => ? = 3
[(1,4),(2,7),(3,6),(5,8)]
 => [4,7,6,1,8,3,2,5] => [4,1,6,3,7,2,8,5] => [5,8,3,6,2,7,1,4] => ? = 4
[(1,5),(2,7),(3,6),(4,8)]
 => [5,7,6,8,1,3,2,4] => [5,1,6,3,7,2,8,4] => [4,8,3,6,2,7,1,5] => ? = 5
[(1,6),(2,7),(3,5),(4,8)]
 => [6,7,5,8,3,1,2,4] => [5,3,6,1,7,2,8,4] => [4,6,3,8,2,7,1,5] => ? = 4
[(1,7),(2,6),(3,5),(4,8)]
 => [7,6,5,8,3,2,1,4] => [5,3,6,2,7,1,8,4] => [4,6,3,7,2,8,1,5] => ? = 3
[(1,8),(2,6),(3,5),(4,7)]
 => [8,6,5,7,3,2,4,1] => [5,3,6,2,7,4,8,1] => [4,6,3,7,2,5,1,8] => ? = 2
[(1,8),(2,5),(3,6),(4,7)]
 => [8,5,6,7,2,3,4,1] => [5,2,6,3,7,4,8,1] => [4,7,3,6,2,5,1,8] => ? = 3
[(1,7),(2,5),(3,6),(4,8)]
 => [7,5,6,8,2,3,1,4] => [5,2,6,3,7,1,8,4] => [4,7,3,6,2,8,1,5] => ? = 4
[(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

$13\!\!-\!\!2$ .