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Matching statistic: St000319
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(load all 4 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,1] => [1,1]
=> 0
[2] => [2]
=> 1
[1,1,1] => [1,1,1]
=> 0
[1,2] => [2,1]
=> 1
[2,1] => [2,1]
=> 1
[3] => [3]
=> 2
[1,1,1,1] => [1,1,1,1]
=> 0
[1,1,2] => [2,1,1]
=> 1
[1,2,1] => [2,1,1]
=> 1
[1,3] => [3,1]
=> 2
[2,1,1] => [2,1,1]
=> 1
[2,2] => [2,2]
=> 1
[3,1] => [3,1]
=> 2
[4] => [4]
=> 3
[1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,1,1,2] => [2,1,1,1]
=> 1
[1,1,2,1] => [2,1,1,1]
=> 1
[1,1,3] => [3,1,1]
=> 2
[1,2,1,1] => [2,1,1,1]
=> 1
[1,2,2] => [2,2,1]
=> 1
[1,3,1] => [3,1,1]
=> 2
[1,4] => [4,1]
=> 3
[2,1,1,1] => [2,1,1,1]
=> 1
[2,1,2] => [2,2,1]
=> 1
[2,2,1] => [2,2,1]
=> 1
[2,3] => [3,2]
=> 2
[3,1,1] => [3,1,1]
=> 2
[3,2] => [3,2]
=> 2
[4,1] => [4,1]
=> 3
[5] => [5]
=> 4
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
[1,1,1,1,2] => [2,1,1,1,1]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 1
[1,1,1,3] => [3,1,1,1]
=> 2
[1,1,2,1,1] => [2,1,1,1,1]
=> 1
[1,1,2,2] => [2,2,1,1]
=> 1
[1,1,3,1] => [3,1,1,1]
=> 2
[1,1,4] => [4,1,1]
=> 3
[1,2,1,1,1] => [2,1,1,1,1]
=> 1
[1,2,1,2] => [2,2,1,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> 1
[1,2,3] => [3,2,1]
=> 2
[1,3,1,1] => [3,1,1,1]
=> 2
[1,3,2] => [3,2,1]
=> 2
[1,4,1] => [4,1,1]
=> 3
[1,5] => [5,1]
=> 4
[2,1,1,1,1] => [2,1,1,1,1]
=> 1
[2,1,1,2] => [2,2,1,1]
=> 1
[2,1,2,1] => [2,2,1,1]
=> 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
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(load all 4 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,1] => [1,1]
=> 0
[2] => [2]
=> 1
[1,1,1] => [1,1,1]
=> 0
[1,2] => [2,1]
=> 1
[2,1] => [2,1]
=> 1
[3] => [3]
=> 2
[1,1,1,1] => [1,1,1,1]
=> 0
[1,1,2] => [2,1,1]
=> 1
[1,2,1] => [2,1,1]
=> 1
[1,3] => [3,1]
=> 2
[2,1,1] => [2,1,1]
=> 1
[2,2] => [2,2]
=> 1
[3,1] => [3,1]
=> 2
[4] => [4]
=> 3
[1,1,1,1,1] => [1,1,1,1,1]
=> 0
[1,1,1,2] => [2,1,1,1]
=> 1
[1,1,2,1] => [2,1,1,1]
=> 1
[1,1,3] => [3,1,1]
=> 2
[1,2,1,1] => [2,1,1,1]
=> 1
[1,2,2] => [2,2,1]
=> 1
[1,3,1] => [3,1,1]
=> 2
[1,4] => [4,1]
=> 3
[2,1,1,1] => [2,1,1,1]
=> 1
[2,1,2] => [2,2,1]
=> 1
[2,2,1] => [2,2,1]
=> 1
[2,3] => [3,2]
=> 2
[3,1,1] => [3,1,1]
=> 2
[3,2] => [3,2]
=> 2
[4,1] => [4,1]
=> 3
[5] => [5]
=> 4
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0
[1,1,1,1,2] => [2,1,1,1,1]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 1
[1,1,1,3] => [3,1,1,1]
=> 2
[1,1,2,1,1] => [2,1,1,1,1]
=> 1
[1,1,2,2] => [2,2,1,1]
=> 1
[1,1,3,1] => [3,1,1,1]
=> 2
[1,1,4] => [4,1,1]
=> 3
[1,2,1,1,1] => [2,1,1,1,1]
=> 1
[1,2,1,2] => [2,2,1,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> 1
[1,2,3] => [3,2,1]
=> 2
[1,3,1,1] => [3,1,1,1]
=> 2
[1,3,2] => [3,2,1]
=> 2
[1,4,1] => [4,1,1]
=> 3
[1,5] => [5,1]
=> 4
[2,1,1,1,1] => [2,1,1,1,1]
=> 1
[2,1,1,2] => [2,2,1,1]
=> 1
[2,1,2,1] => [2,2,1,1]
=> 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001498
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,1] => [2] => [1,1] => [1,0,1,0]
=> 1
[2] => [1] => [1] => [1,0]
=> ? = 0
[1,1,1] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,2}
[2,1] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,2}
[3] => [1] => [1] => [1,0]
=> ? ∊ {0,1,2}
[1,1,1,1] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,2,3}
[1,3] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,2,3}
[2,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,2] => [2] => [1,1] => [1,0,1,0]
=> 1
[3,1] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,2,3}
[4] => [1] => [1] => [1,0]
=> ? ∊ {0,1,2,3}
[1,1,1,1,1] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,3] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,2,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,3,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,2,2,3,4}
[1,4] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,2,2,3,4}
[2,1,1,1] => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,2,2,3,4}
[2,2,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,2,2,3,4}
[3,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,2] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,2,2,3,4}
[4,1] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,2,2,3,4}
[5] => [1] => [1] => [1,0]
=> ? ∊ {0,1,1,2,2,3,4}
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,2] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,2,2] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,4] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1,1] => [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,2,1,2] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[1,2,2,1] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,2,3] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[1,3,1,1] => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,3,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[1,4,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[1,5] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[2,1,1,1,1] => [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[2,1,1,2] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[2,1,3] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[2,2,1,1] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,2,2] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[2,3,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[2,4] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[3,1,1,1] => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[3,1,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[3,2,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[3,3] => [2] => [1,1] => [1,0,1,0]
=> 1
[4,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[4,2] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[5,1] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[6] => [1] => [1] => [1,0]
=> ? ∊ {0,1,1,1,1,2,2,2,3,3,3,4,4,5}
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,3] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,1,2,2] => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,3,1] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,2,1,2] => [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,2,1] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,2,3] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,3,1,1] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,3,2] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,4,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,5] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 3
[1,2,1,2,1] => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[1,2,1,3] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[1,2,3,1] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[1,2,4] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[1,3,1,2] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[1,3,2,1] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[1,4,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[1,5,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[1,6] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[2,1,3,1] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[2,1,4] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[2,3,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[2,4,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[2,5] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[3,1,2,1] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[3,1,3] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[3,4] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[4,1,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[4,2,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
[4,3] => [1,1] => [2] => [1,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,5,5,6}
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001727
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 70%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 70%
Values
[1] => [1]
=> [1,0,1,0]
=> [2,1] => 0
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[2] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 3
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 3
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 4
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,5}
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 1
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 3
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 1
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 3
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 4
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 1
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
[6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,5}
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? ∊ {0,1,1,1,1,1,1,5,5,6}
[1,1,1,1,1,1,1,1] => [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]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,2,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,2,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,2,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,2,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,2,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,3,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,6,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,7] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[2,1,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[2,1,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[2,1,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[2,1,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[2,1,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[2,2,1,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[3,1,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,5,5,5,5,5,6,6,7}
[1,1,1,1,1,1,1,1,1] => [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]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,8}
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
Matching statistic: St000993
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,2}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,2}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,2}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,2}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,3}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,3}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,3,3,4}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 2
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
[1,1,1,2,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,5] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[1,1,2,1,3] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,4] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,3,2,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001645
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 70%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 70%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1] => [2] => [2] => ([],2)
=> ? = 1 + 1
[2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1] => [3] => [3] => ([],3)
=> ? = 2 + 1
[1,2] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1] => [4] => [4] => ([],4)
=> ? ∊ {1,1,2} + 1
[1,1,2] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,2} + 1
[1,2,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[2,2] => [2] => [2] => ([],2)
=> ? ∊ {1,1,2} + 1
[3,1] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => [5] => ([],5)
=> ? ∊ {1,1,1,2,2,2,4} + 1
[1,1,1,2] => [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,4} + 1
[1,1,2,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,4} + 1
[1,1,3] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,4} + 1
[1,2,1,1] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,4} + 1
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[1,3,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,4} + 1
[2,1,2] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,4} + 1
[2,3] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[3,2] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4,1] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => [6] => ([],6)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,1,1,1,2] => [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,1,1,2,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,1,1,3] => [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,1,2,1,1] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,1,2,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,1,3,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,1,4] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,2,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,2,1,2] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,2,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,2,3] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,1,1] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[1,3,2] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,1,1,1] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[2,1,1,2] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[2,1,2,1] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,1,3] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,2,1,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[2,2,2] => [3] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[2,3,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,1,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[3,1,2] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,3] => [2] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5} + 1
[4,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[4,2] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[5,1] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => [7] => ([],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,1,1,1,2] => [5,1] => [1,5] => ([(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,1,1,2,1] => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,1,1,3] => [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,1,2,1,1] => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,1,2,2] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,1,3,1] => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,1,4] => [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,2,1,1,1] => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,2,1,2] => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,2,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,2,3] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,3,1,1] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,3,2] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,4,1] => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,1,5] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,2,1,1,1,1] => [1,1,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,2,1,1,2] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,2,1,2,1] => [1,1,1,1,1] => [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)
=> 5 = 4 + 1
[1,2,1,3] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,2,1,1] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,2,2,2] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,2,3,1] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,2,4] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6} + 1
[1,3,1,2] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,3,2,1] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 4 = 3 + 1
[1,4,2] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,5,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,6] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1,2,1,1] => [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[2,1,3,1] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,1,4] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,3,2] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,4,1] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,5] => [1,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000777
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 70%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 70%
Values
[1] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1] => [2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,2} + 1
[2,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,2} + 1
[3] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,2] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,3} + 1
[1,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,2,3} + 1
[1,3] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,2,3} + 1
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,2] => [2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,2,3} + 1
[4] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => [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)
=> 2 = 1 + 1
[1,1,1,2] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[1,1,2,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[1,1,3] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[1,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[1,4] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[2,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[2,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[2,3] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[4,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,2,3,3,4} + 1
[5] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => [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)
=> 2 = 1 + 1
[1,1,1,1,2] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,1,1,2,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,1,1,3] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,1,2,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,3,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,1,4] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,2,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,2,1,2] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,2,2,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,2,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,4,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[1,5] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[2,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[2,1,2,1] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[2,1,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[2,2,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,2,2] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[2,4] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[3,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[3,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[3,3] => [2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[5,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4,4,5} + 1
[6] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => [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)
=> 2 = 1 + 1
[1,1,1,1,1,2] => [5,1] => [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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,1,3] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,1,2,2] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,3,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,4] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,2,1,1,1] => [2,1,3] => [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)
=> 4 = 3 + 1
[1,1,2,1,2] => [2,1,1,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,2,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,2,3] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,3,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,3,2] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,4,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,5] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,2,1,1,1,1] => [1,1,4] => [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)
=> 3 = 2 + 1
[1,2,1,1,2] => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,2,1,2,1] => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,2,1,3] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,2,2,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,2,2,2] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,4,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,1,1,1,1,1] => [1,5] => [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)
=> 3 = 2 + 1
[2,1,2,1,1] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,1,2,2] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,2,1,1,1] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[7] => [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1,1,1,2,1,1] => [4,1,2] => [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)
=> 4 = 3 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000454
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 60%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[2] => [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1] => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,2] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3] => [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1,1] => [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,3} + 1
[1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,3} + 1
[2,2] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,2,3} + 1
[3,1] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4] => [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1,1,1] => [5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,2,2,3,3,4} + 1
[1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,3,3,4} + 1
[1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,3,3,4} + 1
[1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,3,3,4} + 1
[1,2,2] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,3,3,4} + 1
[1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,2,2,3,3,4} + 1
[2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,3,3,4} + 1
[2,3] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,2,2,3,3,4} + 1
[3,2] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,1] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[5] => [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1,1,1,1] => [6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[2,2,2] => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,3] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[4,1,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4,5} + 1
[4,2] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,1] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[6] => [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1,1,1,1,1,1] => [7] => ([],7)
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,1,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,1,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,2,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,2,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,1,4] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,2,1,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,2,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,2,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,3,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,4,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,1,5] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,2,1,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,2,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,2,1,2,1] => [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)
=> ([(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
[1,2,1,3] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,2,1,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,2,2,2] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,2,3,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,2,4] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,3,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,6} + 1
[1,3,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,4,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,5,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,6] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,3,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,1,4] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,5] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
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: St000013
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 4 = 3 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4} + 1
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6} + 1
[1,1,1,2,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6} + 1
[1,1,1,2,1,3] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6} + 1
[1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6} + 1
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6} + 1
[1,1,1,2,4] => [1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6} + 1
[1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6} + 1
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St001879
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 0
[1,1] => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1}
[2] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1}
[1,1,1] => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,2] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1}
[3] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,1,2] => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,1,1,1,1,2}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,3] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,2}
[2,1,1] => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {0,1,1,1,1,2}
[2,2] => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,2}
[3,1] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,2}
[4] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,1,1] => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[1,2,2] => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,4] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[2,2,1] => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[2,3] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[3,1,1] => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[3,2] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[4,1] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[5] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,3,3}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,1,4] => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,5] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[2,2,2] => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[2,4] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[3,3] => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[4,1,1] => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[4,2] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[5,1] => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[6] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5}
[1,2,1,2,1] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,4] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,4,2] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,5,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[2,1,3,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,1,4] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[2,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[2,4,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[3,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,1,3] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[4,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[4,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,2,1,3,1] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,2,1,4] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,3,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,4,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,2,5] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,2,1] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,3,1,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,3,4] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,4,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,4,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[1,4,3] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,5,2] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[1,6,1] => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2
[2,1,2,1,2] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,1,2,3] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,1,3,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
The following 106 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000932The number of occurrences of the pattern UDU in a Dyck path. St000393The number of strictly increasing runs in a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St001733The number of weak left to right maxima of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000075The orbit size of a standard tableau under promotion. St000442The maximal area to the right of an up step of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001949The rigidity index of a graph. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000782The indicator function of whether a given perfect matching is an L & P matching. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000662The staircase size of the code of a permutation. St000237The number of small exceedances. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. St001668The number of points of the poset minus the width of the poset. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000306The bounce count of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000259The diameter of a connected graph. St001330The hat guessing number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001274The number of indecomposable injective modules with projective dimension equal to two. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001118The acyclic chromatic index of a graph. St000028The number of stack-sorts needed to sort a permutation. St000141The maximum drop size of a permutation. St000451The length of the longest pattern of the form k 1 2. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000652The maximal difference between successive positions of a permutation. St001717The largest size of an interval in a poset. St001271The competition number of a graph. St001644The dimension of a graph. St001060The distinguishing index of a graph. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000120The number of left tunnels of a Dyck path. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001481The minimal height of a peak of a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000741The Colin de Verdière graph invariant. St000456The monochromatic index of a connected graph. St001589The nesting number of a perfect matching. St000317The cycle descent number of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000080The rank of the poset. St000155The number of exceedances (also excedences) of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001530The depth of a Dyck path. St001590The crossing number of a perfect matching. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001414Half the length of the longest odd length palindromic prefix of a binary word. St000455The second largest eigenvalue of a graph if it is integral.
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