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Your data matches 445 different statistics following compositions of up to 3 maps.
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Matching statistic: St000810
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000810: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000810: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [2,2]
=> 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> [3,3]
=> 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> [2,2]
=> 3
Description
The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions.
For example, $p_{22} = 2m_{22} + m_4$, so the statistic on the partition $22$ is 3.
Matching statistic: St001564
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001564: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001564: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2]
=> 10
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [2,2]
=> 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [2,2]
=> 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0]
=> [3,3]
=> 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,0]
=> [3,3]
=> 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> [2,2]
=> 3
Description
The value of the forgotten symmetric functions when all variables set to 1.
Let $f_\lambda(x)$ denote the forgotten symmetric functions.
Then the statistic associated with $\lambda$, where $\lambda$ has $\ell$ parts,
is $f_\lambda(1,1,\dotsc,1)$ where there are $\ell$ variables substituted by $1$.
Matching statistic: St000100
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 10 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 10 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,9),(2,10),(3,8),(4,7),(5,3),(5,10),(6,4),(7,2),(7,5),(8,9),(10,1),(10,8)],11)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> 1 = 3 - 2
Description
The number of linear extensions of a poset.
Matching statistic: St001330
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 10 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 10 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 10 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 3 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 10 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 10 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ?
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ?
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ?
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ?
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ([(0,3),(0,7),(1,9),(3,8),(4,6),(4,10),(5,4),(5,12),(6,1),(6,11),(7,5),(7,8),(8,12),(9,2),(10,11),(11,9),(12,10)],13)
=> ?
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ([(0,7),(2,8),(3,11),(4,5),(4,8),(5,6),(5,10),(6,3),(6,9),(7,2),(7,4),(8,10),(9,11),(10,9),(11,1)],12)
=> ?
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ?
=> ?
=> ? = 3 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ?
=> ?
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ([(0,9),(1,8),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8),(7,9)],10)
=> ? = 3 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ?
=> ?
=> ? = 3 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ?
=> ?
=> ? = 3 - 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000267
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000267: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000267: Graphs ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8 = 10 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8 = 10 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 8 = 10 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 8 = 10 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 8 = 10 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [2,7,4,1,3,5,6] => ([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 8 = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [2,3,7,1,4,5,8,6] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> ? = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,3,8,1,4,7,5,6] => ([(0,7),(1,5),(2,5),(3,6),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [2,3,8,1,6,4,5,7] => ([(0,7),(1,5),(2,5),(3,6),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [2,3,8,5,1,4,6,7] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [2,3,4,8,1,5,6,9,7] => ?
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,4,7,1,5,9,6,8] => ?
=> ? = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,4,6,1,9,5,7,8] => ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,1,0,0]
=> [2,3,4,9,1,5,8,6,7] => ([(0,8),(1,7),(2,7),(3,7),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0,1,0]
=> [2,3,4,9,1,7,5,6,8] => ?
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [2,3,4,9,6,1,5,7,8] => ([(0,8),(1,8),(2,7),(3,7),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,5,9,1,6,7,8] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(5,7),(6,7),(7,8)],9)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,8,1,6,10,7,9] => ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,5,6,10,1,7,8,9] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,8),(6,8),(7,8),(8,9)],10)
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,9,1,6,7,10,8] => ?
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,10,1,7,8,11,9] => ?
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,4,5,6,9,1,7,11,8,10] => ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,11,1,8,9,10] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,9),(7,9),(8,9),(9,10)],11)
=> ? = 3 - 2
Description
The number of maximal spanning forests contained in a graph.
A maximal spanning forest in a graph is a maximal acyclic subgraph. In other words, a spanning forest is a union of spanning trees in all connected components. See also [1] for this and further definitions.
For connected graphs, this is the same as [[St000096]].
Matching statistic: St000264
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [4] => [1] => ([],1)
=> ? = 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => ([(0,1)],2)
=> ? = 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? = 3
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? = 3
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? = 10
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? = 10
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? = 10
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [1,2] => ([(1,2)],3)
=> ? = 10
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1,1] => [1,2] => ([(1,2)],3)
=> ? = 10
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [4,1,1] => [1,2] => ([(1,2)],3)
=> ? = 10
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,1,1] => [1,2] => ([(1,2)],3)
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,4,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 10
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [4,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 10
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [4,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 10
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,4,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,3,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [3,2,1,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [4,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ? = 10
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [4,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ? = 10
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [4,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ? = 10
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1] => [1,4] => ([(3,4)],5)
=> ? = 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [2,3,1,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1,1] => [1,5] => ([(4,5)],6)
=> ? = 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,4,1,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,4,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [2,3,1,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [4,1,1,1,1,1,1] => [1,6] => ([(5,6)],7)
=> ? = 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001644
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 10 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 10 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 10 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 10 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 10 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 2 = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? = 3 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,6] => ([(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,6] => ([(5,8),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [4,6] => ([(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,7] => ([(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3 - 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,7] => ([(6,9),(7,9),(8,9)],10)
=> ? = 3 - 1
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St000298
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 10 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 10 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,9),(2,10),(3,8),(4,7),(5,3),(5,10),(6,4),(7,2),(7,5),(8,9),(10,1),(10,8)],11)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 3 - 2
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000307
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 10 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 10 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,9),(2,10),(3,8),(4,7),(5,3),(5,10),(6,4),(7,2),(7,5),(8,9),(10,1),(10,8)],11)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 3 - 2
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000845
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 50%
Values
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 10 - 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 10 - 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,9),(2,10),(3,8),(4,7),(5,3),(5,10),(6,4),(7,2),(7,5),(8,9),(10,1),(10,8)],11)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,6),(1,9),(3,8),(4,7),(5,1),(5,8),(6,4),(7,3),(7,5),(8,9),(9,2)],10)
=> ? = 10 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 3 - 2
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 3 - 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ?
=> ? = 3 - 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10)
=> ? = 3 - 2
Description
The maximal number of elements covered by an element in a poset.
The following 435 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000846The maximal number of elements covering an element of a poset. St000632The jump number of the poset. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000455The second largest eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000618The number of self-evacuating tableaux of given shape. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000667The greatest common divisor of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000741The Colin de Verdière graph invariant. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000908The length of the shortest maximal antichain in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000993The multiplicity of the largest part of an integer partition. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001268The size of the largest ordinal summand in the poset. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001389The number of partitions of the same length below the given integer partition. St001399The distinguishing number of a poset. St001432The order dimension of the partition. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001779The order of promotion on the set of linear extensions of a poset. St001780The order of promotion on the set of standard tableaux of given shape. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001881The number of factors of a lattice as a Cartesian product of lattices. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000929The constant term of the character polynomial of an integer partition. St000944The 3-degree of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001301The first Betti number of the order complex associated with the poset. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001625The Möbius invariant of a lattice. St001657The number of twos in an integer partition. St001845The number of join irreducibles minus the rank of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001902The number of potential covers of a poset. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000806The semiperimeter of the associated bargraph. St000096The number of spanning trees of a graph. St001081The number of minimal length factorizations of a permutation into star transpositions. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001735The number of permutations with the same set of runs. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. 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. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000454The largest eigenvalue of a graph if it is integral. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001820The size of the image of the pop stack sorting operator. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001846The number of elements which do not have a complement in the lattice. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000075The orbit size of a standard tableau under promotion. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001060The distinguishing index of a graph. St001875The number of simple modules with projective dimension at most 1. 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. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000933The number of multipartitions of sizes given by an integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000145The Dyson rank of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000228The size of a partition. St000260The radius of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000108The number of partitions contained in the given partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000567The sum of the products of all pairs of parts. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000877The depth of the binary word interpreted as a path. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001618The cardinality of the Frattini sublattice of a lattice. St001668The number of points of the poset minus the width of the poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St000326The position of the first one in a binary word after appending a 1 at the end. St000519The largest length of a factor maximising the subword complexity. St000630The length of the shortest palindromic decomposition of a binary word. St000842The breadth of a permutation. St000862The number of parts of the shifted shape of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000922The minimal number such that all substrings of this length are unique. St000945The number of matchings in the dihedral orbit of a perfect matching. St001394The genus of a permutation. St001488The number of corners of a skew partition. St001545The second Elser number of a connected graph. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St000035The number of left outer peaks of a permutation. St000295The length of the border of a binary word. St000308The height of the tree associated to a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000617The number of global maxima of a Dyck path. St000627The exponent of a binary word. St000703The number of deficiencies of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000883The number of longest increasing subsequences of a permutation. St001260The permanent of an alternating sign matrix. St001626The number of maximal proper sublattices of a lattice. St001895The oddness of a signed permutation. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000297The number of leading ones in a binary word. St000356The number of occurrences of the pattern 13-2. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000629The defect of a binary word. St000731The number of double exceedences of a permutation. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000871The number of very big ascents of a permutation. St000878The number of ones minus the number of zeros of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001703The villainy of a graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001848The atomic length of a signed permutation. St001851The number of Hecke atoms of a signed permutation. St001862The number of crossings of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000735The last entry on the main diagonal of a standard tableau. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000874The position of the last double rise in a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St000977MacMahon's equal index of a Dyck path. St000978The sum of the positions of double down-steps of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000984The number of boxes below precisely one peak. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000047The number of standard immaculate tableaux of a given shape. St000352The Elizalde-Pak rank of a permutation. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000662The staircase size of the code of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000058The order of a permutation. St000141The maximum drop size of a permutation. St000259The diameter of a connected graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000402Half the size of the symmetry class of a permutation. St000422The energy of a graph, if it is integral. St000665The number of rafts of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000834The number of right outer peaks of a permutation. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St001372The length of a longest cyclic run of ones of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001645The pebbling number of a connected graph. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000054The first entry of the permutation. St000068The number of minimal elements in a poset. St000296The length of the symmetric border of a binary word. St000651The maximal size of a rise in a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000742The number of big ascents of a permutation after prepending zero. St001090The number of pop-stack-sorts needed to sort a permutation. St001118The acyclic chromatic index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000065The number of entries equal to -1 in an alternating sign matrix. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000189The number of elements in the poset. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000218The number of occurrences of the pattern 213 in a permutation. St000223The number of nestings in the permutation. St000237The number of small exceedances. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000393The number of strictly increasing runs in a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St001817The number of flag weak exceedances of a signed permutation. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001865The number of alignments of a signed permutation. St000302The determinant of the distance matrix of a connected graph. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000456The monochromatic index of a connected graph. St000381The largest part of an integer composition. St000480The number of lower covers of a partition in dominance order. St000546The number of global descents of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001812The biclique partition number of a graph. St000159The number of distinct parts of the integer partition. St000383The last part of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001570The minimal number of edges to add to make a graph Hamiltonian. St001597The Frobenius rank of a skew partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000124The cardinality of the preimage of the Simion-Schmidt map. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000680The Grundy value for Hackendot on posets. St000696The number of cycles in the breakpoint graph of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000764The number of strong records in an integer composition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000912The number of maximal antichains in a poset. St000996The number of exclusive left-to-right maxima of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001596The number of two-by-two squares inside a skew partition. St001638The book thickness of a graph. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000288The number of ones in a binary word. St000441The number of successions of a permutation. St000462The major index minus the number of excedences of a permutation. St000477The weight of a partition according to Alladi. St000656The number of cuts of a poset. St000699The toughness times the least common multiple of 1,. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000787The number of flips required to make a perfect matching noncrossing. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001379The number of inversions plus the major index of a permutation. St001434The number of negative sum pairs of a signed permutation. St001519The pinnacle sum of a permutation. St001651The Frankl number of a lattice. St001717The largest size of an interval in a poset. St001947The number of ties in a parking function. St000691The number of changes of a binary word. St001437The flex of a binary word. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St000464The Schultz index of a connected graph. St001706The number of closed sets in a graph.
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