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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St001918
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 2
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 3
[3,1]
=> 2
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 4
[4,1]
=> 3
[3,2]
=> 4
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 5
[5,1]
=> 4
[4,2]
=> 3
[4,1,1]
=> 3
[3,3]
=> 2
[3,2,1]
=> 4
[3,1,1,1]
=> 2
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 6
[6,1]
=> 5
[5,2]
=> 8
[5,1,1]
=> 4
[4,3]
=> 9
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 2
[3,2,2]
=> 4
[3,2,1,1]
=> 4
[3,1,1,1,1]
=> 2
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 7
[7,1]
=> 6
[6,2]
=> 5
[6,1,1]
=> 5
[5,3]
=> 12
[5,2,1]
=> 8
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let λ be an integer partition of n and let N be the least common multiple of the parts of λ. Fix an arbitrary permutation π of cycle type λ. Then π induces a cyclic action of order N on {1,…,n}.
The corresponding character can be identified with the cyclic sieving polynomial Cλ(q) of this action, modulo qN−1. Explicitly, it is
∑p∈λ[p]qN/p,
where [p]q=1+⋯+qp−1 is the q-integer.
This statistic records the degree of Cλ(q). Equivalently, it equals
(1−1λ1)N,
where λ1 is the largest part of λ.
The statistic is undefined for the empty partition.
Matching statistic: St001232
(load all 74 compositions to match this statistic)
(load all 74 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 16%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 16%
Values
[1]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> ? = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,1,4} + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,4} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,4} + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,4} + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,1,4} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,4} + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 5 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,3,4,6,8,9} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,6,8,9} + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,6,8,9} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,6,8,9} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,6,8,9} + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,3,4,6,8,9} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,6,8,9} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,3,4,6,8,9} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,3,4,6,8,9} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6 = 5 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,3,4,4,6,7,8,9,12} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,2,3,3,4,4,5,6,7,8,8,8,9,9,12,12,16} + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001645
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 13%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 13%
Values
[1]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[2]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[1,1]
=> 11 => [2] => ([],2)
=> ? = 1 + 1
[3]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1]
=> 111 => [3] => ([],3)
=> ? = 2 + 1
[4]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[3,1]
=> 11 => [2] => ([],2)
=> ? ∊ {1,1,2,3} + 1
[2,2]
=> 00 => [2] => ([],2)
=> ? ∊ {1,1,2,3} + 1
[2,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,3} + 1
[1,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {1,1,2,3} + 1
[5]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {2,4,4} + 1
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[2,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {2,4,4} + 1
[1,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {2,4,4} + 1
[6]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[5,1]
=> 11 => [2] => ([],2)
=> ? ∊ {1,1,1,2,3,3,4,4,5} + 1
[4,2]
=> 00 => [2] => ([],2)
=> ? ∊ {1,1,1,2,3,3,4,4,5} + 1
[4,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,2,3,3,4,4,5} + 1
[3,3]
=> 11 => [2] => ([],2)
=> ? ∊ {1,1,1,2,3,3,4,4,5} + 1
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {1,1,1,2,3,3,4,4,5} + 1
[2,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {1,1,1,2,3,3,4,4,5} + 1
[2,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,3,3,4,4,5} + 1
[2,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,2,3,3,4,4,5} + 1
[1,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? ∊ {1,1,1,2,3,3,4,4,5} + 1
[7]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[4,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[3,3,1]
=> 111 => [3] => ([],3)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[3,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[3,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[3,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[2,2,1,1,1]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[2,1,1,1,1,1]
=> 011111 => [1,5] => ([(4,5)],6)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[1,1,1,1,1,1,1]
=> 1111111 => [7] => ([],7)
=> ? ∊ {2,2,3,4,4,4,5,6,8,9} + 1
[8]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[7,1]
=> 11 => [2] => ([],2)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[6,2]
=> 00 => [2] => ([],2)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[6,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[5,3]
=> 11 => [2] => ([],2)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[4,4]
=> 00 => [2] => ([],2)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[4,3,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[4,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[4,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[4,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[3,3,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[3,2,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[3,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[2,2,2,2]
=> 0000 => [4] => ([],4)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[2,2,2,1,1]
=> 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[2,2,1,1,1,1]
=> 001111 => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[2,1,1,1,1,1,1]
=> 0111111 => [1,6] => ([(5,6)],7)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[1,1,1,1,1,1,1,1]
=> 11111111 => [8] => ([],8)
=> ? ∊ {1,1,1,1,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[9]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[7,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[6,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[5,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5,3,1]
=> 111 => [3] => ([],3)
=> ? ∊ {2,2,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[10]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[11]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
[10,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[9,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[8,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[7,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6,5]
=> 01 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,4,3]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[12]
=> 0 => [1] => ([],1)
=> 1 = 0 + 1
[9,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[7,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[7,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[6,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[5,5,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[5,4,3]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[13]
=> 1 => [1] => ([],1)
=> 1 = 0 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000771
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 16%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 16%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> ? = 1 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,2} + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,2} + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,2,3} + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,2,3} + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,3} + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,3,4,4} + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {1,1,3,4,4} + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,3,4,4} + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,3,4,4} + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,3,4,4} + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,1,2,3,4,4,5} + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,1,2,3,4,4,5} + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,3,4,4,5} + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,1,2,3,4,4,5} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,3,4,4,5} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,3,4,4,5} + 1
[2,2,2]
=> [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)
=> 3 = 2 + 1
[2,2,1,1]
=> [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)
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,3,4,4,5} + 1
[1,1,1,1,1,1]
=> [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)
=> 4 = 3 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[3,3,1]
=> [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)
=> 1 = 0 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[2,2,2,1]
=> [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)
=> 2 = 1 + 1
[2,2,1,1,1]
=> [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)
=> 3 = 2 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,2,3,3,4,4,5,6,8,9} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 5 = 4 + 1
[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] => ([],8)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 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] => ([(6,7)],8)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[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] => ([(5,7),(6,7)],8)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 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] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[4,4]
=> [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)
=> 2 = 1 + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[3,3,2]
=> [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)
=> 2 = 1 + 1
[3,3,1,1]
=> [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)
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[2,2,2,2]
=> [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)
=> 3 = 2 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,3,3,3,4,4,4,4,5,5,6,7,8,9,12} + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => ([(6,8),(7,8)],9)
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[4,4,1]
=> [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)
=> 2 = 1 + 1
[3,3,3]
=> [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)
=> 4 = 3 + 1
[3,3,2,1]
=> [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)
=> 1 = 0 + 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[5,5]
=> [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)
=> 3 = 2 + 1
[4,4,2]
=> [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)
=> 2 = 1 + 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[3,3,2,2]
=> [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)
=> 2 = 1 + 1
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,3,4,2,6,7,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[4,4,3]
=> [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)
=> 2 = 1 + 1
[4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [6,2,3,5,4,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(0,3),(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)
=> 3 = 2 + 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ([(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)
=> 3 = 2 + 1
[3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,2,4,5,3,7,1] => ([(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)
=> 1 = 0 + 1
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => ([(0,6),(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)
=> 2 = 1 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => ([(0,2),(0,3),(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)
=> 4 = 3 + 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ([(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)
=> 2 = 1 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(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)
=> 5 = 4 + 1
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 1 + 1
[3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ([(0,5),(0,6),(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)
=> 2 = 1 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
(4−1−2−1−14−1−2−2−14−1−1−2−14).
Its eigenvalues are 0,4,4,6, so the statistic is 2.
The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore statistic 1.
Matching statistic: St000777
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> ? = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? ∊ {0,1} + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,1} + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,1,3} + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,1,3} + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,3} + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ? ∊ {0,1,1,4,4} + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,1,1,4,4} + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,1,4,4} + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,1,4,4} + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,4,4} + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,1,1,3,4,4,5} + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,1,1,3,4,4,5} + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,3,4,4,5} + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,1,1,3,4,4,5} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,3,4,4,5} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,3,4,4,5} + 1
[2,2,2]
=> [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)
=> 2 = 1 + 1
[2,2,1,1]
=> [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)
=> 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,3,4,4,5} + 1
[1,1,1,1,1,1]
=> [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)
=> 3 = 2 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[3,3,1]
=> [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)
=> 5 = 4 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[2,2,2,1]
=> [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)
=> 4 = 3 + 1
[2,2,1,1,1]
=> [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)
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,2,4,4,5,6,8,9} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[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] => ([],8)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 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] => ([(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[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] => ([(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 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] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[4,4]
=> [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)
=> 3 = 2 + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[3,3,2]
=> [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)
=> 4 = 3 + 1
[3,3,1,1]
=> [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)
=> 5 = 4 + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[2,2,2,2]
=> [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)
=> 3 = 2 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,4,4,4,5,5,6,7,8,9,12} + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => ([(6,8),(7,8)],9)
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,2,2,2,3,3,3,4,4,5,5,6,7,8,8,8,9,9,12,12,16} + 1
[4,4,1]
=> [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)
=> 5 = 4 + 1
[3,3,3]
=> [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)
=> 2 = 1 + 1
[3,3,2,1]
=> [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)
=> 6 = 5 + 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[5,5]
=> [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)
=> 3 = 2 + 1
[4,4,2]
=> [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)
=> 4 = 3 + 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[3,3,2,2]
=> [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)
=> 5 = 4 + 1
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,3,4,2,6,7,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[4,4,3]
=> [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)
=> 4 = 3 + 1
[4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [6,2,3,5,4,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(0,3),(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)
=> 4 = 3 + 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ([(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)
=> 4 = 3 + 1
[3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,2,4,5,3,7,1] => ([(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)
=> 7 = 6 + 1
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => ([(0,6),(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)
=> 5 = 4 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => ([(0,2),(0,3),(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)
=> 3 = 2 + 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ([(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)
=> 5 = 4 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(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)
=> 2 = 1 + 1
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ([(0,5),(0,6),(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)
=> 5 = 4 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 18%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> ? = 0
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> ? = 0
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? ∊ {0,2}
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,2}
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,2,3}
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,2,3}
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,3}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ? ∊ {0,2,3,4,4}
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,2,3,4,4}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,2,3,4,4}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,3,4,4}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,3,4,4}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,2,3,3,4,4,5}
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,2,3,3,4,4,5}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,3,3,4,4,5}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,3,3,4,4,5}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,2,3,3,4,4,5}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,3,4,4,5}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,2,3,3,4,4,5}
[1,1,1,1,1,1]
=> [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)
=> 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[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] => ([],8)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[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] => ([(6,7)],8)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => ([(4,6),(5,6)],7)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[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] => ([(5,7),(6,7)],8)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => ([(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => ([(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,3] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,3,3,4,4,4,4,5,5,6,7,8,9,12}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? ∊ {0,1,1,2,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? ∊ {0,1,1,2,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,6,7] => ([(5,7),(6,7)],8)
=> ? ∊ {0,1,1,2,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => ([(6,8),(7,8)],9)
=> ? ∊ {0,1,1,2,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 2
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [3,5,1,2,6,7,4] => ([(0,6),(1,6),(2,3),(2,4),(3,5),(4,5),(5,6)],7)
=> 2
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 5
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [4,5,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [3,4,6,1,2,7,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 3
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1
[3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [3,5,6,1,2,7,4] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 4
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 5
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 5
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [3,4,5,7,1,2,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [3,6,1,2,4,7,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> 2
[4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 5
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [5,1,6,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,5,6,7,1,2,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 6
[2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001491
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 5%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 5%
Values
[1]
=> []
=> => ? => ? = 0
[2]
=> []
=> => ? => ? ∊ {0,1}
[1,1]
=> [1]
=> 10 => 00 => ? ∊ {0,1}
[3]
=> []
=> => ? => ? ∊ {0,2}
[2,1]
=> [1]
=> 10 => 00 => ? ∊ {0,2}
[1,1,1]
=> [1,1]
=> 110 => 010 => 1
[4]
=> []
=> => ? => ? ∊ {0,1,3}
[3,1]
=> [1]
=> 10 => 00 => ? ∊ {0,1,3}
[2,2]
=> [2]
=> 100 => 000 => ? ∊ {0,1,3}
[2,1,1]
=> [1,1]
=> 110 => 010 => 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[5]
=> []
=> => ? => ? ∊ {0,3,4,4}
[4,1]
=> [1]
=> 10 => 00 => ? ∊ {0,3,4,4}
[3,2]
=> [2]
=> 100 => 000 => ? ∊ {0,3,4,4}
[3,1,1]
=> [1,1]
=> 110 => 010 => 1
[2,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? ∊ {0,3,4,4}
[6]
=> []
=> => ? => ? ∊ {0,2,3,3,4,4,5}
[5,1]
=> [1]
=> 10 => 00 => ? ∊ {0,2,3,3,4,4,5}
[4,2]
=> [2]
=> 100 => 000 => ? ∊ {0,2,3,3,4,4,5}
[4,1,1]
=> [1,1]
=> 110 => 010 => 1
[3,3]
=> [3]
=> 1000 => 0000 => ? ∊ {0,2,3,3,4,4,5}
[3,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[2,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[2,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? ∊ {0,2,3,3,4,4,5}
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? ∊ {0,2,3,3,4,4,5}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? ∊ {0,2,3,3,4,4,5}
[7]
=> []
=> => ? => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[6,1]
=> [1]
=> 10 => 00 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[5,2]
=> [2]
=> 100 => 000 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[5,1,1]
=> [1,1]
=> 110 => 010 => 1
[4,3]
=> [3]
=> 1000 => 0000 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[4,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[3,3,1]
=> [3,1]
=> 10010 => 00010 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[3,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[3,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[2,2,2,1]
=> [2,2,1]
=> 11010 => 01010 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 001110 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111110 => ? ∊ {0,2,3,3,4,4,4,5,6,8,9}
[8]
=> []
=> => ? => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[7,1]
=> [1]
=> 10 => 00 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[6,2]
=> [2]
=> 100 => 000 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[6,1,1]
=> [1,1]
=> 110 => 010 => 1
[5,3]
=> [3]
=> 1000 => 0000 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[5,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[4,4]
=> [4]
=> 10000 => 00000 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[4,3,1]
=> [3,1]
=> 10010 => 00010 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[4,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[4,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[3,3,2]
=> [3,2]
=> 10100 => 00100 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[3,3,1,1]
=> [3,1,1]
=> 100110 => 000110 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[3,2,2,1]
=> [2,2,1]
=> 11010 => 01010 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 001110 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[2,2,2,2]
=> [2,2,2]
=> 11100 => 01100 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 010110 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0011110 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111110 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 01111110 => ? ∊ {0,1,2,3,3,3,3,4,4,4,4,5,5,6,7,8,9,12}
[9]
=> []
=> => ? => ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[8,1]
=> [1]
=> 10 => 00 => ? ∊ {0,1,2,2,3,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[7,1,1]
=> [1,1]
=> 110 => 010 => 1
[6,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[6,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[5,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[8,1,1]
=> [1,1]
=> 110 => 010 => 1
[7,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[7,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[6,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[9,1,1]
=> [1,1]
=> 110 => 010 => 1
[8,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[8,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[7,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[10,1,1]
=> [1,1]
=> 110 => 010 => 1
[9,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[9,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[8,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[11,1,1]
=> [1,1]
=> 110 => 010 => 1
[10,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[10,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[9,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[12,1,1]
=> [1,1]
=> 110 => 010 => 1
[11,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[11,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[10,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[13,1,1]
=> [1,1]
=> 110 => 010 => 1
[12,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[12,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[11,2,2]
=> [2,2]
=> 1100 => 0100 => 1
[14,1,1]
=> [1,1]
=> 110 => 010 => 1
[13,2,1]
=> [2,1]
=> 1010 => 0010 => 1
[13,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2
[12,2,2]
=> [2,2]
=> 1100 => 0100 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let An=K[x]/(xn).
We associate to a nonempty subset S of an (n-1)-set the module MS, which is the direct sum of An-modules with indecomposable non-projective direct summands of dimension i when i is in S (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of MS. We decode the subset as a binary word so that for example the subset S={1,3} of {1,2,3} is decoded as 101.
Matching statistic: St000306
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 24%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 24%
Values
[1]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[4]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[3,1]
=> 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,2]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[5]
=> 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[4,1]
=> 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 4 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[6]
=> 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[5,1]
=> 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,2]
=> 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4} + 1
[3,3]
=> 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,3,4} + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4} + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[7]
=> 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[6,1]
=> 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[5,2]
=> 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {1,2,3,4,4,8,9} + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,4,4,8,9} + 1
[4,3]
=> 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,2,3,4,4,8,9} + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,4,4,8,9} + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,2,3,4,4,8,9} + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,4,4,8,9} + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,2,3,4,4,8,9} + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[8]
=> 100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[7,1]
=> 100000010 => [1,7,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[6,2]
=> 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[5,3]
=> 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[5,1,1,1]
=> 100001110 => [1,5,1,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[4,4]
=> 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[4,3,1]
=> 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[4,2,2]
=> 1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[4,2,1,1]
=> 10010110 => [1,3,2,1,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[4,1,1,1,1]
=> 100011110 => [1,4,1,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[3,3,2]
=> 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[3,3,1,1]
=> 1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[3,2,2,1]
=> 1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[3,2,1,1,1]
=> 10101110 => [1,2,2,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[3,1,1,1,1,1]
=> 100111110 => [1,3,1,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[2,2,2,2]
=> 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[2,2,2,1,1]
=> 1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[2,2,1,1,1,1]
=> 11011110 => [1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[2,1,1,1,1,1,1]
=> 101111110 => [1,2,1,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3,3,3,3,4,4,4,5,5,6,8,9,12} + 1
[1,1,1,1,1,1,1,1]
=> 111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[9]
=> 1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[8,1]
=> 1000000010 => [1,8,2] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[7,2]
=> 100000100 => [1,6,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[7,1,1]
=> 1000000110 => [1,7,1,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[6,3]
=> 10001000 => [1,4,4] => [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[6,2,1]
=> 100001010 => [1,5,2,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[6,1,1,1]
=> 1000001110 => [1,6,1,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[5,4]
=> 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[5,3,1]
=> 10010010 => [1,3,3,2] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[5,2,2]
=> 10001100 => [1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[5,2,1,1]
=> 100010110 => [1,4,2,1,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[5,1,1,1,1]
=> 1000011110 => [1,5,1,1,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[4,4,1]
=> 1100010 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[4,3,2]
=> 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[4,3,1,1]
=> 10100110 => [1,2,3,1,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[4,2,2,1]
=> 10011010 => [1,3,1,2,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[4,2,1,1,1]
=> 100101110 => [1,3,2,1,1,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[4,1,1,1,1,1]
=> 1000111110 => [1,4,1,1,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[3,3,3]
=> 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[3,3,2,1]
=> 1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[3,3,1,1,1]
=> 11001110 => [1,1,3,1,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[3,2,2,2]
=> 1011100 => [1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[3,2,2,1,1]
=> 10110110 => [1,2,1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[3,2,1,1,1,1]
=> 101011110 => [1,2,2,1,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,8,8,9,9,12,12,16} + 1
[2,2,2,2,1]
=> 1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 9 = 8 + 1
[5,5]
=> 1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
[6,5]
=> 10100000 => [1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 2 = 1 + 1
[2,2,2,2,2,1]
=> 11111010 => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[6,6]
=> 11000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 2 = 1 + 1
[7,7]
=> 110000000 => [1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> 2 = 1 + 1
Description
The bounce count of a Dyck path.
For a Dyck path D of length 2n, this is the number of points (i,i) for 1≤i<n that are touching points of the [[Mp00099|bounce path]] of D.
Matching statistic: St000454
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 18%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 18%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 0
[2]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1
[1,1]
=> [1,1,0,0]
=> [2] => ([],2)
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ? = 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,4}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {0,1,2,4}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,4}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,4}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(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)
=> 5
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,3,4}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,3,4}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,3,4}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => ([(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)
=> ? ∊ {1,1,1,2,3,4}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => ([(0,3),(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)
=> ? ∊ {1,1,1,2,3,4}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => ([(0,2),(0,3),(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)
=> ? ∊ {1,1,1,2,3,4}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,2,4,4,4,8,9}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,2,4,4,4,8,9}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,2,4,4,4,8,9}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,2,4,4,4,8,9}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,2,4,4,4,8,9}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,2,4,4,4,8,9}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,2,4,4,4,8,9}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,2,4,4,4,8,9}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,2,4,4,4,8,9}
[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] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,3,1] => ([(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)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,1,1,1,1,3,3,3,4,4,4,5,5,6,7,8,9,12}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,2,1] => ([(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,3,1] => ([(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)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,2,1,1] => ([(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,5,6,7,8,8,8,9,9,12,12,16}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 0
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> 1
[5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => ([],6)
=> 0
[5,5,3]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[4,3,3,3]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> 1
[4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => ([],7)
=> 0
Description
The largest eigenvalue of a graph if it is integral.
If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001875
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 5%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 5%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> ([],1)
=> ? = 0
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,1}
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ? ∊ {0,1}
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,2}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,2}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ? ∊ {0,1,2}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,2,3}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,2,3}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,2,3}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,2,3}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,2,3}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,2,3,4,4}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,2,3,4,4}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,2,3,4,4}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,2,3,4,4}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,2,3,4,4}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,2,3,4,4}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,2,3,4,4}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,5}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,3,3,4,4,4,5,6,8,9}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,5,5,6,7,8,9,12}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,5,5,6,7,8,9,12}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,5,5,6,7,8,9,12}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,5,5,6,7,8,9,12}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,5,5,6,7,8,9,12}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,2,2,3,3,4,4,4,4,5,5,6,7,8,9,12}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 3
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,4,2,2]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[5,4,4,2,1]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> 3
[5,4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of simple modules with projective dimension at most 1.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St000392The length of the longest run of ones in a binary word. St000422The energy of a graph, if it is integral. St000982The length of the longest constant subword. St001330The hat guessing number of a graph. St000628The balance of a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000259The diameter of a connected graph. St001638The book thickness of a graph. St001644The dimension of a graph. St001488The number of corners of a skew partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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