Your data matches 17 different statistics following compositions of up to 3 maps.
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Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> 0
[1,-2] => [1]
=> [1,0]
=> 0
[-1,2] => [1]
=> [1,0]
=> 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> 1
[2,-1] => [2]
=> [1,0,1,0]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> 0
[1,2,-3] => [1]
=> [1,0]
=> 0
[1,-2,3] => [1]
=> [1,0]
=> 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> 0
[-1,3,2] => [1]
=> [1,0]
=> 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,-3,-2] => [1]
=> [1,0]
=> 0
[2,1,-3] => [1]
=> [1,0]
=> 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,-1,-3] => [1]
=> [1,0]
=> 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,-2,-1] => [1]
=> [1,0]
=> 0
[1,2,3,-4] => [1]
=> [1,0]
=> 0
[1,2,-3,4] => [1]
=> [1,0]
=> 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[-1,2,3,4] => [1]
=> [1,0]
=> 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
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: St000041
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
St000041: Perfect matchings ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[1,-2] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[-1,2] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[2,-1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[1,2,-3] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[1,-2,3] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[-1,3,2] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-1,-3,-2] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[2,1,-3] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-2,-1,-3] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[-3,-2,-1] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[1,2,3,-4] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[1,2,-3,4] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
[-1,2,3,4] => [1]
=> [1,0]
=> [(1,2)]
=> 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 2
Description
The number of nestings of a perfect matching. This is the number of pairs of edges $((a,b), (c,d))$ such that $a\le c\le d\le b$. i.e., the edge $(c,d)$ is nested inside $(a,b)$.
Matching statistic: St000161
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00029: Dyck paths to binary tree: left tree, up step, right tree, down stepBinary trees
St000161: Binary trees ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [.,.]
=> 0
[1,-2] => [1]
=> [1,0]
=> [.,.]
=> 0
[-1,2] => [1]
=> [1,0]
=> [.,.]
=> 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[2,-1] => [2]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[1,2,-3] => [1]
=> [1,0]
=> [.,.]
=> 0
[1,-2,3] => [1]
=> [1,0]
=> [.,.]
=> 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> [.,.]
=> 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[-1,3,2] => [1]
=> [1,0]
=> [.,.]
=> 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
[-1,-3,-2] => [1]
=> [1,0]
=> [.,.]
=> 0
[2,1,-3] => [1]
=> [1,0]
=> [.,.]
=> 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
[-2,-1,-3] => [1]
=> [1,0]
=> [.,.]
=> 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> [.,.]
=> 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
[-3,-2,-1] => [1]
=> [1,0]
=> [.,.]
=> 0
[1,2,3,-4] => [1]
=> [1,0]
=> [.,.]
=> 0
[1,2,-3,4] => [1]
=> [1,0]
=> [.,.]
=> 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> [.,.]
=> 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 2
[-1,2,3,4] => [1]
=> [1,0]
=> [.,.]
=> 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [.,[.,.]]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 2
[3,4,5,6,7,1,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[3,4,6,1,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[4,1,5,6,7,3,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[4,1,6,3,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[5,6,2,7,4,1,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[6,1,2,-7,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[6,1,7,-5,2,3,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[7,-5,-4,-6,1,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[-2,-5,-7,1,3,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[7,-6,2,1,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[-3,-5,7,-6,1,2,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[5,4,1,-7,2,3,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[7,4,1,-6,2,3,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[-5,4,1,-7,-6,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[3,1,6,2,4,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[4,6,1,2,3,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[6,1,5,2,4,8,7,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[6,5,1,2,3,8,7,-4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[1,7,5,3,8,4,6,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[1,7,6,8,3,4,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[1,5,8,3,7,4,6,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[1,6,8,7,3,4,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[-7,-3,2,8,-5,1,4,6] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> ? = 4
[-8,-7,-5,4,-6,-2,1,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[6,7,8,3,5,-4,1,2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ? = 0
[7,8,-3,-6,2,4,-5,1] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> ? = 4
[2,8,-7,-6,-5,4,1,3] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> ? = 4
[-2,-6,-3,-8,-7,5,1,4] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [[[[.,.],.],.],[[.,[.,.]],.]]
=> ? = 4
Description
The sum of the sizes of the right subtrees of a binary tree. This statistic corresponds to [[St000012]] under the Tamari Dyck path-binary tree bijection, and to [[St000018]] of the $312$-avoiding permutation corresponding to the binary tree. It is also the sum of all heights $j$ of the coordinates $(i,j)$ of the Dyck path corresponding to the binary tree.
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
St000018: Permutations ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1] => 0
[1,-2] => [1]
=> [1,0]
=> [1] => 0
[-1,2] => [1]
=> [1,0]
=> [1] => 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[-2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[1,2,-3] => [1]
=> [1,0]
=> [1] => 0
[1,-2,3] => [1]
=> [1,0]
=> [1] => 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,3] => [1]
=> [1,0]
=> [1] => 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[-1,3,2] => [1]
=> [1,0]
=> [1] => 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-1,-3,-2] => [1]
=> [1,0]
=> [1] => 0
[2,1,-3] => [1]
=> [1,0]
=> [1] => 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-2,-1,-3] => [1]
=> [1,0]
=> [1] => 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[3,-2,1] => [1]
=> [1,0]
=> [1] => 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-3,-2,-1] => [1]
=> [1,0]
=> [1] => 0
[1,2,3,-4] => [1]
=> [1,0]
=> [1] => 0
[1,2,-3,4] => [1]
=> [1,0]
=> [1] => 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1] => 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1] => 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-7,-3,2,8,-5,1,4,6] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ? = 4
[-8,-3,2,6,-7,-5,1,4] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-8,-6,-3,-7,-5,1,2,4] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ? = 2
[5,6,2,3,-4,1,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[5,6,1,-3,2,4,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[7,8,-3,-6,2,4,-5,1] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ? = 4
[7,8,2,3,-6,5,-4,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[2,8,-7,-6,-5,4,1,3] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ? = 4
[7,8,1,-3,-6,5,2,4] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[4,5,2,3,-6,1,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-5,4,1,-3,-6,2,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-2,-3,-4,-5,-6,1,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-2,-3,-4,-7,6,-5,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-2,-8,1,-6,-7,4,3,5] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-6,-2,-3,1,7,8,4,5] => [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ? = 2
[-2,-6,-3,-8,-7,5,1,4] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ? = 4
[-2,-5,4,-3,-6,-7,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[5,7,1,6,2,-4,-8,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-2,1,-7,6,3,-5,-8,4] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[4,-5,-8,-6,-7,2,1,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-2,-6,1,-8,-7,5,3,4] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[-2,1,-4,-5,-6,-7,-8,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
[4,-5,-6,-8,-7,3,1,2] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ? = 3
Description
The number of inversions of a permutation. This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Matching statistic: St000246
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000246: Permutations ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1] => 0
[1,-2] => [1]
=> [1,0]
=> [1] => 0
[-1,2] => [1]
=> [1,0]
=> [1] => 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [2,1] => 0
[-2,1] => [2]
=> [1,0,1,0]
=> [2,1] => 0
[1,2,-3] => [1]
=> [1,0]
=> [1] => 0
[1,-2,3] => [1]
=> [1,0]
=> [1] => 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,2,3] => [1]
=> [1,0]
=> [1] => 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [2,1] => 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [2,1] => 0
[-1,3,2] => [1]
=> [1,0]
=> [1] => 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[-1,-3,-2] => [1]
=> [1,0]
=> [1] => 0
[2,1,-3] => [1]
=> [1,0]
=> [1] => 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> [2,1] => 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [2,1] => 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[-2,-1,-3] => [1]
=> [1,0]
=> [1] => 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [2,1] => 0
[3,-2,1] => [1]
=> [1,0]
=> [1] => 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [2,1] => 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[-3,-2,-1] => [1]
=> [1,0]
=> [1] => 0
[1,2,3,-4] => [1]
=> [1,0]
=> [1] => 0
[1,2,-3,4] => [1]
=> [1,0]
=> [1] => 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1] => 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1] => 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[-7,4,-3,1,2,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[-8,-7,2,4,-5,-3,1,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[-8,4,6,-7,-5,-2,1,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[-8,6,3,-7,-5,-4,1,2] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[-7,-3,2,8,-5,1,4,6] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 4
[-8,-3,2,6,-7,-5,1,4] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-8,6,2,-7,-5,-4,1,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[7,8,-6,-4,5,1,2,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[5,8,2,3,6,-4,-7,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[5,6,2,3,-4,1,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[5,6,1,-3,2,4,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[3,6,-8,-5,1,-4,-7,2] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[7,8,-3,-6,2,4,-5,1] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 4
[-6,-2,1,3,7,8,4,5] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[7,8,2,3,-6,5,-4,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[2,-8,6,-3,1,-5,-7,4] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[5,8,6,1,2,-4,-7,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[6,-8,1,-3,2,-5,-7,4] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[7,8,4,2,3,-6,-5,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[2,8,-7,-6,-5,4,1,3] => [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 4
[7,8,5,-4,-6,1,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[7,8,1,-3,-6,5,2,4] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[4,5,2,3,-6,1,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-5,4,1,-3,-6,2,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-2,-3,-4,-5,-6,1,-8,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-2,-3,-4,-7,6,-5,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 3
[-3,8,-7,-4,1,-5,2,6] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
[-5,4,-3,1,6,-7,-8,2] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,4,3,2,1] => ? = 1
Description
The number of non-inversions of a permutation. For a permutation of $\{1,\ldots,n\}$, this is given by $\operatorname{noninv}(\pi) = \binom{n}{2}-\operatorname{inv}(\pi)$.
Matching statistic: St001558
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
St001558: Permutations ⟶ ℤResult quality: 98% values known / values provided: 98%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1] => 0
[1,-2] => [1]
=> [1,0]
=> [1] => 0
[-1,2] => [1]
=> [1,0]
=> [1] => 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[-2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[1,2,-3] => [1]
=> [1,0]
=> [1] => 0
[1,-2,3] => [1]
=> [1,0]
=> [1] => 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,3] => [1]
=> [1,0]
=> [1] => 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[-1,3,2] => [1]
=> [1,0]
=> [1] => 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-1,-3,-2] => [1]
=> [1,0]
=> [1] => 0
[2,1,-3] => [1]
=> [1,0]
=> [1] => 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-2,-1,-3] => [1]
=> [1,0]
=> [1] => 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[3,-2,1] => [1]
=> [1,0]
=> [1] => 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[-3,-2,-1] => [1]
=> [1,0]
=> [1] => 0
[1,2,3,-4] => [1]
=> [1,0]
=> [1] => 0
[1,2,-3,4] => [1]
=> [1,0]
=> [1] => 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1] => 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1] => 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [2,1] => 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ? = 1
[3,1,5,2,8,4,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,5,1,2,8,3,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[7,4,1,8,2,3,5,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,1,5,2,7,4,6,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,5,1,2,7,3,6,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,4,1,6,2,3,5,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,6,7,1,2,3,4,-5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,6,1,7,2,3,5,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,1,6,7,2,4,5,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[5,6,8,1,2,3,4,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[6,7,8,1,2,3,4,-5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[6,1,7,8,2,4,5,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[5,1,7,2,8,4,6,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[5,7,1,2,8,3,6,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,6,1,8,2,3,5,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[3,1,6,8,2,4,5,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,1,2,7,8,5,6,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[3,1,8,2,7,4,6,-5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[3,1,8,7,2,4,5,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,8,1,7,2,3,5,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,8,1,2,7,3,6,-5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[5,8,7,1,2,3,4,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[6,8,5,1,2,3,4,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[7,8,1,6,2,3,5,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,7,5,1,2,3,4,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,5,2,3,8,1,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[7,5,1,3,8,2,6,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,5,6,3,4,1,2,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,5,7,3,1,4,2,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,6,7,1,3,4,2,-5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,6,1,7,3,5,2,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[8,1,6,7,4,5,2,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[7,1,6,8,4,2,5,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[7,5,8,3,1,2,4,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[5,6,8,2,3,1,4,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,7,2,8,1,3,5,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,6,2,8,3,1,5,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[4,8,2,7,1,5,3,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[6,8,1,7,2,5,3,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
[5,8,6,2,4,1,3,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ? = 0
Description
The number of transpositions that are smaller or equal to a permutation in Bruhat order. A statistic is known to be '''smooth''' if and only if this number coincides with the number of inversions. This is also equivalent for a permutation to avoid the two pattern $4231$ and $3412$.
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St001295: Dyck paths ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> 0
[1,-2] => [1]
=> [1,0]
=> 0
[-1,2] => [1]
=> [1,0]
=> 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> 1
[2,-1] => [2]
=> [1,0,1,0]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> 0
[1,2,-3] => [1]
=> [1,0]
=> 0
[1,-2,3] => [1]
=> [1,0]
=> 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> 0
[-1,3,2] => [1]
=> [1,0]
=> 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-1,-3,-2] => [1]
=> [1,0]
=> 0
[2,1,-3] => [1]
=> [1,0]
=> 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-2,-1,-3] => [1]
=> [1,0]
=> 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[-3,-2,-1] => [1]
=> [1,0]
=> 0
[1,2,3,-4] => [1]
=> [1,0]
=> 0
[1,2,-3,4] => [1]
=> [1,0]
=> 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[-1,2,3,4] => [1]
=> [1,0]
=> 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[3,4,5,6,7,1,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,4,6,1,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,1,5,6,7,3,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,1,6,3,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,6,2,7,4,1,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,1,2,-7,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[-7,4,-3,1,2,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[6,1,7,-5,2,3,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,-5,-4,-6,1,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[-2,-5,-7,1,3,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,-6,2,1,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[-3,-5,7,-6,1,2,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[5,4,1,-7,2,3,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,4,1,-6,2,3,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[-5,4,1,-7,-6,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,1,6,2,4,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,1,5,2,8,4,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,5,1,2,8,3,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[4,6,1,2,3,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,1,5,2,4,8,7,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6,5,1,2,3,8,7,-4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[7,4,1,8,2,3,5,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,7,5,3,8,4,6,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,1,5,2,7,4,6,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,5,1,2,7,3,6,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,4,1,6,2,3,5,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,6,7,1,2,3,4,-5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8,6,1,7,2,3,5,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
Description
Gives the vector space dimension of the homomorphism space between J^2 and J^2.
Matching statistic: St000005
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00030: Dyck paths zeta mapDyck paths
St000005: Dyck paths ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-2] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-2,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-1,3,2] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-3,-2] => [1]
=> [1,0]
=> [1,0]
=> 0
[2,1,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-2,-1,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> [1,0]
=> 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-3,-2,-1] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,2,3,-4] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,2,-3,4] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[3,4,5,6,7,1,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,6,1,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,1,5,6,7,3,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,1,6,3,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,6,2,7,4,1,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[6,1,2,-7,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[-7,4,-3,1,2,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[6,1,7,-5,2,3,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,-5,-4,-6,1,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[-2,-5,-7,1,3,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,-6,2,1,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[-3,-5,7,-6,1,2,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,4,1,-7,2,3,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,4,1,-6,2,3,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[-5,4,1,-7,-6,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,1,6,2,4,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,1,5,2,8,4,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,5,1,2,8,3,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,6,1,2,3,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[6,1,5,2,4,8,7,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[6,5,1,2,3,8,7,-4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,4,1,8,2,3,5,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,7,5,3,8,4,6,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[8,1,5,2,7,4,6,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,5,1,2,7,3,6,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,4,1,6,2,3,5,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,6,7,1,2,3,4,-5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,6,1,7,2,3,5,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The bounce statistic of a Dyck path. The '''bounce path''' $D'$ of a Dyck path $D$ is the Dyck path obtained from $D$ by starting at the end point $(2n,0)$, traveling north-west until hitting $D$, then bouncing back south-west to the $x$-axis, and repeating this procedure until finally reaching the point $(0,0)$. The points where $D'$ touches the $x$-axis are called '''bounce points''', and a bounce path is uniquely determined by its bounce points. This statistic is given by the sum of all $i$ for which the bounce path $D'$ of $D$ touches the $x$-axis at $(2i,0)$. In particular, the bounce statistics of $D$ and $D'$ coincide.
Matching statistic: St000006
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St000006: Dyck paths ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-2] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-2,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-1,3,2] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-1,-3,-2] => [1]
=> [1,0]
=> [1,0]
=> 0
[2,1,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-2,-1,-3] => [1]
=> [1,0]
=> [1,0]
=> 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> [1,0]
=> 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-3,-2,-1] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,2,3,-4] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,2,-3,4] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[-1,2,3,4] => [1]
=> [1,0]
=> [1,0]
=> 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[3,4,5,6,7,1,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,4,6,1,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,1,5,6,7,3,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[4,1,6,3,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,6,2,7,4,1,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[6,1,2,-7,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[-7,4,-3,1,2,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[6,1,7,-5,2,3,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,-5,-4,-6,1,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[-2,-5,-7,1,3,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,-6,2,1,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[-3,-5,7,-6,1,2,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[5,4,1,-7,2,3,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,4,1,-6,2,3,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[-5,4,1,-7,-6,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,1,6,2,4,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,1,5,2,8,4,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,5,1,2,8,3,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,6,1,2,3,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[6,1,5,2,4,8,7,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[6,5,1,2,3,8,7,-4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[7,4,1,8,2,3,5,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,7,5,3,8,4,6,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[8,1,5,2,7,4,6,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,5,1,2,7,3,6,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,4,1,6,2,3,5,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,6,7,1,2,3,4,-5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,6,1,7,2,3,5,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The dinv of a Dyck path. Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see [[St000012]]). The dinv statistic of $D$ is $$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$ Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose ''arm length'' is one larger or equal to its ''leg length''. There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2]. Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by $$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$
Matching statistic: St000057
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00033: Dyck paths to two-row standard tableauStandard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[-1] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,-2] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[-1,2] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[2,-1] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[-2,1] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,2,-3] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,-2,3] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[-1,2,3] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 2
[1,3,-2] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[1,-3,2] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[-1,3,2] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 1
[-1,-3,-2] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[2,1,-3] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[2,-1,3] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 1
[-2,1,3] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 1
[-2,-1,-3] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[3,2,-1] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[3,-2,1] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 1
[-3,2,1] => [2]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> 0
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 1
[-3,-2,-1] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,2,3,-4] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,2,-3,4] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,-2,3,4] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 2
[-1,2,3,4] => [1]
=> [1,0]
=> [[1],[2]]
=> 0
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [[1,2],[3,4]]
=> 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 2
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
=> ? = 1
[4,5,2,3,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[2,-8,-4,-6,-5,1,3,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 3
[-4,3,-5,2,-8,-6,1,7] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 3
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 3
[-4,3,5,8,7,-6,1,2] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 3
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 3
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 3
[3,-7,2,8,-5,1,4,6] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[-7,6,-8,-4,-2,1,3,5] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[5,-6,-7,-4,2,3,-8,1] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[-8,-6,5,-4,2,-7,1,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[-7,6,1,-8,-5,-4,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 3
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
=> ? = 1
[4,-8,-5,-7,1,-6,2,3] => [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,13,14],[2,4,6,8,10,12,15,16]]
=> ? = 1
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]]
=> ? = 3
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
=> ? = 1
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
=> ? = 1
[3,4,5,6,7,1,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[3,4,6,1,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[4,1,5,6,7,3,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[4,1,6,3,7,5,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[5,6,2,7,4,1,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[6,1,2,-7,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[-7,4,-3,1,2,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]]
=> ? = 1
[6,1,7,-5,2,3,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[7,-5,-4,-6,1,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[-2,-5,-7,1,3,4,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[7,-6,2,1,3,4,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[-3,-5,7,-6,1,2,4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[5,4,1,-7,2,3,6] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[7,4,1,-6,2,3,5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[-5,4,1,-7,-6,2,3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[3,1,6,2,4,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[3,1,5,2,8,4,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 0
[4,5,1,2,8,3,6,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 0
[4,6,1,2,3,8,7,-5] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[6,1,5,2,4,8,7,-3] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[6,5,1,2,3,8,7,-4] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[7,4,1,8,2,3,5,-6] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 0
[1,7,5,3,8,4,6,-2] => [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
=> ? = 0
[8,1,5,2,7,4,6,-3] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 0
[8,5,1,2,7,3,6,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 0
[8,4,1,6,2,3,5,-7] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 0
[8,6,7,1,2,3,4,-5] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 0
[8,6,1,7,2,3,5,-4] => [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
=> ? = 0
Description
The Shynar inversion number of a standard tableau. Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000947The major index east count of a Dyck path. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.