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Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St001228
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
St001228: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 3
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> 5
[1,1,1,0,0,0]
=> 6
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 5
[1,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,0]
=> 6
[1,0,1,1,1,0,0,0]
=> 7
[1,1,0,0,1,0,1,0]
=> 5
[1,1,0,0,1,1,0,0]
=> 6
[1,1,0,1,0,0,1,0]
=> 6
[1,1,0,1,0,1,0,0]
=> 7
[1,1,0,1,1,0,0,0]
=> 8
[1,1,1,0,0,0,1,0]
=> 7
[1,1,1,0,0,1,0,0]
=> 8
[1,1,1,0,1,0,0,0]
=> 9
[1,1,1,1,0,0,0,0]
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> 11
[1,1,0,0,1,0,1,0,1,0]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> 11
[1,1,0,1,1,1,0,0,0,0]
=> 12
Description
The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000012
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 6
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 8
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 11
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 11
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 12
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000400
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000400: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000400: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[]]
=> 1
[1,0,1,0]
=> [[],[]]
=> 2
[1,1,0,0]
=> [[[]]]
=> 3
[1,0,1,0,1,0]
=> [[],[],[]]
=> 3
[1,0,1,1,0,0]
=> [[],[[]]]
=> 4
[1,1,0,0,1,0]
=> [[[]],[]]
=> 4
[1,1,0,1,0,0]
=> [[[],[]]]
=> 5
[1,1,1,0,0,0]
=> [[[[]]]]
=> 6
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 4
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 5
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 5
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 6
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 7
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 5
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 6
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 6
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 7
[1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 8
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 7
[1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 8
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 9
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 11
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 11
[1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 12
Description
The path length of an ordered tree.
This is the sum of the lengths of all paths from the root to a node, see Section 2.3.4.5 of [1].
Matching statistic: St000041
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 5
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 6
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> 7
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 6
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> 7
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 8
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> 7
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> 8
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 9
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10)]
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9)]
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [(1,12),(2,3),(4,7),(5,6),(8,9),(10,11)]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [(1,12),(2,3),(4,7),(5,6),(8,11),(9,10)]
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)]
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,6),(7,10),(8,9)]
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)]
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)]
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)]
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)]
=> 11
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [(1,12),(2,5),(3,4),(6,7),(8,11),(9,10)]
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [(1,12),(2,5),(3,4),(6,9),(7,8),(10,11)]
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [(1,12),(2,5),(3,4),(6,11),(7,8),(9,10)]
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [(1,12),(2,5),(3,4),(6,11),(7,10),(8,9)]
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,9),(10,11)]
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [(1,12),(2,7),(3,4),(5,6),(8,11),(9,10)]
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [(1,12),(2,9),(3,4),(5,6),(7,8),(10,11)]
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)]
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,6),(7,10),(8,9)]
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,4),(5,8),(6,7),(10,11)]
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [(1,12),(2,11),(3,4),(5,8),(6,7),(9,10)]
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)]
=> 11
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)]
=> 12
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: St000231
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000231: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000231: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> {{1}}
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 3
[1,1,0,0]
=> [1,1,0,0]
=> {{1,2}}
=> 2
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 6
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 5
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 4
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 10
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 7
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 8
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 7
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 9
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 6
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 8
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 6
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 5
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 7
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 6
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 5
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 15
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 11
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 12
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 11
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 13
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 9
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 12
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 9
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 8
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 10
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 8
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 14
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 10
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 11
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 10
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 7
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 13
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 9
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 11
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 7
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 10
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 9
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 7
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 6
Description
Sum of the maximal elements of the blocks of a set partition.
Matching statistic: St000947
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 5
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 7
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 9
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 9
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 12
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 11
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 9
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 13
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 11
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 8
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 9
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 13
Description
The major index east count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
Matching statistic: St001161
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 5
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 6
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 7
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 9
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 6
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 9
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 8
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 12
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 11
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 9
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 13
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 11
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 8
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 8
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 9
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 13
Description
The major index north count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index north count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.
Matching statistic: St000639
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 95% ●values known / values provided: 99%●distinct values known / distinct values provided: 95%
Mp00065: Permutations —permutation poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 95% ●values known / values provided: 99%●distinct values known / distinct values provided: 95%
Values
[1,0]
=> [1] => ([],1)
=> ? = 1
[1,0,1,0]
=> [2,1] => ([],2)
=> 2
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 3
[1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 4
[1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> 4
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 5
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 6
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> 5
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 5
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 6
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 7
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> 5
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 6
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 8
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 7
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 8
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 9
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 11
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 11
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 12
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 8
Description
The number of relations in a poset.
This is the number of intervals $x,y$ with $x\leq y$ in the poset, and therefore the dimension of the posets incidence algebra.
Matching statistic: St000081
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 5
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 6
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 7
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 9
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 11
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,6,4,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 11
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 12
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [2,4,3,5,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,3,6,5,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,3,7,5,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,4,3,7,6,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,3,4,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [2,5,4,3,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [2,7,4,3,6,5,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,7,6,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,2,4,7,5,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [3,2,6,4,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,2,7,4,5,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,7,4,6,5,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,7,5,4,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,2,7,5,6,4,1] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,2,3,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [4,2,3,6,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,2,3,7,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,2,3,7,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,2,3,4,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,2,4,3,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [7,2,4,3,6,5,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,3,2,5,7,6,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [4,3,2,6,5,7,1] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,3,2,7,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [5,3,2,4,7,6,1] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [7,3,2,4,6,5,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [6,3,2,5,4,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [7,3,2,5,4,6,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [7,3,2,5,6,4,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,3,4,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [7,3,4,2,6,5,1] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [7,4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,17}
Description
The number of edges of a graph.
Matching statistic: St001397
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2,1] => ([],2)
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([],3)
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 5
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([],4)
=> 6
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> 5
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> 5
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> 6
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> 7
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 6
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 6
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 7
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 8
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 7
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 8
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 9
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([],5)
=> 10
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => ([(1,4),(4,5),(5,2),(5,3)],6)
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => ([(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,6,4,5,1] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> 7
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> 8
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,5,6,1] => ([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> 6
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,3,6,5,1] => ([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 7
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 7
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,6,3,4,5,1] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> 8
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6,3,5,4,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> 9
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,4,3,6,1] => ([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> 8
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,6,4,3,5,1] => ([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> 9
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,6,4,5,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> 10
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6)
=> 11
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => ([(1,5),(2,5),(3,4),(5,3)],6)
=> 6
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,2,4,6,5,1] => ([(1,5),(2,5),(5,3),(5,4)],6)
=> 7
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,6,1] => ([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 7
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,6,4,5,1] => ([(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> 8
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 9
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> 7
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,2,3,6,5,1] => ([(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> 8
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> 8
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6)
=> 9
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => ([(2,3),(3,4),(3,5)],6)
=> 10
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 9
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => ([(2,3),(2,4),(3,5),(4,5)],6)
=> 10
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => ([(2,3),(2,4),(4,5)],6)
=> 11
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(2,3),(2,4),(2,5)],6)
=> 12
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,7,6,1] => ([(1,5),(4,6),(5,4),(6,2),(6,3)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,7,5,6,1] => ([(1,5),(4,3),(5,6),(6,2),(6,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,5,4,6,7,1] => ([(1,5),(2,6),(3,6),(5,2),(5,3),(6,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [2,3,5,4,7,6,1] => ([(1,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,6,4,5,7,1] => ([(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,7,4,5,6,1] => ([(1,6),(4,5),(5,3),(6,2),(6,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,7,4,6,5,1] => ([(1,6),(5,3),(5,4),(6,2),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,4,7,1] => ([(1,5),(2,6),(3,6),(4,6),(5,2),(5,3),(5,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,7,5,4,6,1] => ([(1,5),(3,6),(4,6),(5,2),(5,3),(5,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,7,5,6,4,1] => ([(1,6),(5,4),(6,2),(6,3),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,1] => ([(1,3),(1,4),(3,6),(4,6),(5,2),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,3,4,7,6,1] => ([(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,2)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [2,6,3,4,5,7,1] => ([(1,3),(1,5),(2,6),(3,6),(4,2),(5,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,7,3,4,5,6,1] => ([(1,3),(1,6),(4,5),(5,2),(6,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,7,3,4,6,5,1] => ([(1,4),(1,5),(5,6),(6,2),(6,3)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,6,3,5,4,7,1] => ([(1,4),(1,5),(2,6),(3,6),(4,6),(5,2),(5,3)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,7,3,5,4,6,1] => ([(1,2),(1,5),(3,6),(4,6),(5,3),(5,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,7,3,5,6,4,1] => ([(1,4),(1,6),(5,3),(6,2),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,7,5,4,6,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,2,7,5,6,4,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(6,3)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,2,3,5,7,6,1] => ([(1,6),(2,3),(3,6),(6,4),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [4,2,3,6,5,7,1] => ([(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,2,3,7,5,6,1] => ([(1,5),(1,6),(2,4),(4,5),(4,6),(6,3)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,2,3,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => ([(1,6),(2,3),(3,5),(5,6),(6,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,2,3,4,7,6,1] => ([(1,5),(1,6),(2,3),(3,4),(4,5),(4,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [6,2,3,5,4,7,1] => ([(1,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [7,2,3,5,4,6,1] => ([(2,3),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => ([(1,6),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,2,4,3,7,6,1] => ([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [6,2,4,3,5,7,1] => ([(1,5),(2,3),(2,4),(3,6),(4,6),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,2,4,5,3,7,1] => ([(1,6),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [7,3,2,6,5,4,1] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [6,3,4,2,5,7,1] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [7,3,4,2,5,6,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [7,3,4,2,6,5,1] => ([(2,5),(2,6),(3,4),(4,5),(4,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [7,3,4,5,2,6,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [7,3,5,4,2,6,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [7,5,3,4,6,2,1] => ([(3,6),(4,5),(5,6)],7)
=> ? ∊ {6,7,7,7,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,13,13,14,14,14,15,15,17}
Description
Number of pairs of incomparable elements in a finite poset.
For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000018The number of inversions of a permutation. St000378The diagonal inversion number of an integer partition. St000246The number of non-inversions of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000301The number of facets of the stable set polytope of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000133The "bounce" of a permutation. St000795The mad of a permutation. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000004The major index of a permutation. St000042The number of crossings of a perfect matching. St000224The sorting index of a permutation. St000233The number of nestings of a set partition. St000305The inverse major index of a permutation. St000332The positive inversions of an alternating sign matrix. St000496The rcs statistic of a set partition. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St001428The number of B-inversions of a signed permutation. St001671Haglund's hag of a permutation. St001759The Rajchgot index of a permutation. St001718The number of non-empty open intervals in a poset. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice.
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