Your data matches 51 different statistics following compositions of up to 3 maps.
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Matching statistic: St000460
(load all 2 compositions to match this statistic)
Mapping
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => 2
[1,1]
 => 2
[3]
 => 3
[2,1]
 => 3
[1,1,1]
 => 3
[4]
 => 4
[3,1]
 => 4
[2,2]
 => 1
[2,1,1]
 => 4
[1,1,1,1]
 => 4
[5]
 => 5
[4,1]
 => 5
[3,2]
 => 1
[3,1,1]
 => 5
[2,2,1]
 => 1
[2,1,1,1]
 => 5
[1,1,1,1,1]
 => 5
[6]
 => 6
[5,1]
 => 6
[4,2]
 => 1
[4,1,1]
 => 6
[3,3]
 => 2
[3,2,1]
 => 1
[3,1,1,1]
 => 6
[2,2,2]
 => 2
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 6
[1,1,1,1,1,1]
 => 6
[7]
 => 7
[6,1]
 => 7
[5,2]
 => 1
[5,1,1]
 => 7
[4,3]
 => 2
[4,2,1]
 => 1
[4,1,1,1]
 => 7
[3,3,1]
 => 2
[3,2,2]
 => 2
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 7
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 7
[1,1,1,1,1,1,1]
 => 7
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000461
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => [1,0,1,0]
 => [1,2] => [1,2] => 2
[1,1]
 => [1,1,0,0]
 => [2,1] => [1,2] => 2
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,2,3] => 3
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => [1,2,3] => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3,4] => 4
[2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => [1,3,2] => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,2,3,4] => 4
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,2,3,4] => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4,5] => 5
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,2,4,3] => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,3,4,5] => 5
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,3,4,2] => 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,2,3,4,5] => 5
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,5,6] => 6
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,3,5,4] => 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,4,5,6] => 6
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,4,2,3] => 2
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,2,4,5,3] => 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,3,4,5,6] => 6
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,4,2,3] => 2
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [1,3,4,5,2] => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [1,2,3,4,5,6] => 6
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 6
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => 7
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [1,2,3,4,6,5] => 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => 7
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,2,5,3,4] => 2
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [1,2,3,5,6,4] => 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => 7
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,4,5,2,3] => 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,2,5,3,4] => 2
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => [1,2,4,5,6,3] => 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => 7
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [1,4,5,2,3] => 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => [1,3,4,5,6,2] => 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => [1,2,3,4,5,6,7] => 7
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 7
Description
The rix statistic of a permutation. This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then $rix(w) := 0$ if $i = 1 < k$, $rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and $rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
Matching statistic: St000989
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => [1,0,1,0]
 => [1,2] => [1,2] => 1 = 2 - 1
[1,1]
 => [1,1,0,0]
 => [2,1] => [1,2] => 1 = 2 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 2 = 3 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,2,3] => 2 = 3 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => [1,2,3] => 2 = 3 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3,4] => 3 = 4 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => [1,3,2] => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,2,3,4] => 3 = 4 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,2,3,4] => 3 = 4 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4,5] => 4 = 5 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,2,4,3] => 0 = 1 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,3,4,5] => 4 = 5 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,3,4,2] => 0 = 1 - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,2,3,4,5] => 4 = 5 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 4 = 5 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 5 = 6 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,5,6] => 5 = 6 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,3,5,4] => 0 = 1 - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,4,5,6] => 5 = 6 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,2,4,5,3] => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,3,4,5,6] => 5 = 6 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,4,2,3] => 1 = 2 - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [1,3,4,5,2] => 0 = 1 - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [1,2,3,4,5,6] => 5 = 6 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 5 = 6 - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => [1,2,3,4,6,5] => 0 = 1 - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [1,2,5,3,4] => 1 = 2 - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,4,6,3] => [1,2,3,5,6,4] => 0 = 1 - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,4,5,2,3] => 1 = 2 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,2,5,3,4] => 1 = 2 - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,3,5,6,2] => [1,2,4,5,6,3] => 0 = 1 - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [1,4,5,2,3] => 1 = 2 - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,2,4,5,6,1] => [1,3,4,5,6,2] => 0 = 1 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => [1,2,3,4,5,6,7] => 6 = 7 - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 6 = 7 - 1
Description
The number of final rises of a permutation. For a permutation $\pi$ of length $n$, this is the maximal $k$ such that $$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$ Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St001880
(load all 12 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 71%
Values
[2]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? ∊ {2,2}
[1,1]
 => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? ∊ {2,2}
[3]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3
[2,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1]
 => [1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3
[4]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[2,2]
 => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {1,1}
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1}
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {1,1,1,2,2}
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,2,2}
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,1,1,2,2}
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,2,2}
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,2,2}
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? ∊ {1,1,1,1,2,2,2,2}
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[.,[[[.,.],.],.]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,1,1,1,2,2,2,2}
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? ∊ {1,1,1,1,2,2,2,2}
[4,1,1,1]
 => [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)
 => 7
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,2,2,2}
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,1,1,1,2,2,2,2}
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {1,1,1,1,2,2,2,2}
[3,1,1,1,1]
 => [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)
 => 7
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,2,2,2}
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[.,.],[[[[.,.],.],.],.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? ∊ {1,1,1,1,2,2,2,2}
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 71%
Values
[2]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? ∊ {2,2} - 1
[1,1]
 => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? ∊ {2,2} - 1
[3]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {1,1} - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1} - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 5 - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {1,1,1,2,2} - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,2,2} - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,1,1,2,2} - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,2,2} - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,2,2} - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 6 - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[.,[[.,.],.]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? ∊ {1,1,1,1,2,2,2,2} - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[.,[[[.,.],.],.]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,1,1,1,2,2,2,2} - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? ∊ {1,1,1,1,2,2,2,2} - 1
[4,1,1,1]
 => [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)
 => 6 = 7 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,2,2,2} - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {1,1,1,1,2,2,2,2} - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? ∊ {1,1,1,1,2,2,2,2} - 1
[3,1,1,1,1]
 => [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)
 => 6 = 7 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,2,2,2,2} - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[.,.],[[[[.,.],.],.],.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? ∊ {1,1,1,1,2,2,2,2} - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 7 - 1
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.
Matching statistic: St001772
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001772: Signed permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 86%
Values
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 2
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => 3
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1,4] => 4
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 4
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,3,4,2] => 4
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [1,5,4,3,2] => 4
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? ∊ {1,1,5}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => [3,4,2,1,5] => ? ∊ {1,1,5}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => 5
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2,4] => 5
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 5
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => [1,4,5,3,2] => 5
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => [1,6,5,4,3,2] => ? ∊ {1,1,5}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? ∊ {1,1,1,2,2,6,6,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [4,5,3,2,1,6] => [4,5,3,2,1,6] => ? ∊ {1,1,1,2,2,6,6,6}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [3,2,4,1,5] => ? ∊ {1,1,1,2,2,6,6,6}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {1,1,1,2,2,6,6,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [3,2,1,5,4] => ? ∊ {1,1,1,2,2,6,6,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 6
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [1,4,3,5,2] => 6
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,1,5,4,3] => ? ∊ {1,1,1,2,2,6,6,6}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,3,5,4,2] => 6
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5,6,4,3,2] => [1,5,6,4,3,2] => ? ∊ {1,1,1,2,2,6,6,6}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? ∊ {1,1,1,2,2,6,6,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [5,6,4,3,2,1,7] => [5,6,4,3,2,1,7] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [4,3,5,2,1,6] => [4,3,5,2,1,6] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [3,5,4,2,1,6] => [3,5,4,2,1,6] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [1,4,3,2,5] => 7
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,1,5,4] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,4,5,3] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,3,4,5,2] => 7
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5,4,6,3,2] => [1,5,4,6,3,2] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [1,2,5,4,3] => 7
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,4,6,5,3,2] => [1,4,6,5,3,2] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6,7,5,4,3,2] => [1,6,7,5,4,3,2] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,2,2,2,2,7,7,7,7}
Description
The number of occurrences of the signed pattern 12 in a signed permutation. This is the number of pairs $1\leq i < j\leq n$ such that $0 < \pi(i) < \pi(j)$.
Matching statistic: St000870
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 37%●distinct values known / distinct values provided: 29%
Values
[2]
 => []
 => []
 => ?
 => ? ∊ {2,2}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {2,2}
[3]
 => []
 => []
 => ?
 => ? ∊ {3,3,3}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {3,3,3}
[1,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {3,3,3}
[4]
 => []
 => []
 => ?
 => ? ∊ {4,4,4,4}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {4,4,4,4}
[2,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[2,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {4,4,4,4}
[1,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {4,4,4,4}
[5]
 => []
 => []
 => ?
 => ? ∊ {5,5,5,5,5}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {5,5,5,5,5}
[3,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[3,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {5,5,5,5,5}
[2,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {5,5,5,5,5}
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {5,5,5,5,5}
[6]
 => []
 => []
 => ?
 => ? ∊ {6,6,6,6,6,6}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {6,6,6,6,6,6}
[4,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[4,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {6,6,6,6,6,6}
[3,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[3,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {6,6,6,6,6,6}
[2,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {6,6,6,6,6,6}
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {6,6,6,6,6,6}
[7]
 => []
 => []
 => ?
 => ? ∊ {7,7,7,7,7,7,7}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[5,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[5,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[4,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[4,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
[3,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
Description
The product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001360
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001360: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 37%●distinct values known / distinct values provided: 29%
Values
[2]
 => []
 => []
 => ?
 => ? ∊ {2,2}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {2,2}
[3]
 => []
 => []
 => ?
 => ? ∊ {3,3,3}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {3,3,3}
[1,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {3,3,3}
[4]
 => []
 => []
 => ?
 => ? ∊ {4,4,4,4}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {4,4,4,4}
[2,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[2,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {4,4,4,4}
[1,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {4,4,4,4}
[5]
 => []
 => []
 => ?
 => ? ∊ {5,5,5,5,5}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {5,5,5,5,5}
[3,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[3,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {5,5,5,5,5}
[2,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {5,5,5,5,5}
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {5,5,5,5,5}
[6]
 => []
 => []
 => ?
 => ? ∊ {6,6,6,6,6,6}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {6,6,6,6,6,6}
[4,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[4,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {6,6,6,6,6,6}
[3,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[3,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {6,6,6,6,6,6}
[2,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {6,6,6,6,6,6}
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {6,6,6,6,6,6}
[7]
 => []
 => []
 => ?
 => ? ∊ {7,7,7,7,7,7,7}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[5,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[5,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[4,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[4,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
[3,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
Description
The number of covering relations in Young's lattice below a partition.
Matching statistic: St001378
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001378: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 37%●distinct values known / distinct values provided: 29%
Values
[2]
 => []
 => []
 => ?
 => ? ∊ {2,2}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {2,2}
[3]
 => []
 => []
 => ?
 => ? ∊ {3,3,3}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {3,3,3}
[1,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {3,3,3}
[4]
 => []
 => []
 => ?
 => ? ∊ {4,4,4,4}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {4,4,4,4}
[2,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[2,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {4,4,4,4}
[1,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {4,4,4,4}
[5]
 => []
 => []
 => ?
 => ? ∊ {5,5,5,5,5}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {5,5,5,5,5}
[3,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[3,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {5,5,5,5,5}
[2,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {5,5,5,5,5}
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {5,5,5,5,5}
[6]
 => []
 => []
 => ?
 => ? ∊ {6,6,6,6,6,6}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {6,6,6,6,6,6}
[4,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[4,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {6,6,6,6,6,6}
[3,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[3,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {6,6,6,6,6,6}
[2,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {6,6,6,6,6,6}
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {6,6,6,6,6,6}
[7]
 => []
 => []
 => ?
 => ? ∊ {7,7,7,7,7,7,7}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[5,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[5,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[4,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[4,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
[3,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
Description
The product of the cohook lengths of the integer partition. For a cell $c = (i,j)$, the '''cohook length''' of $c$ is $h^*(c) = i+j-1$. This statistic is then $$\prod_{c \in \lambda} h^*(c).$$
Matching statistic: St001380
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001380: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 37%●distinct values known / distinct values provided: 29%
Values
[2]
 => []
 => []
 => ?
 => ? ∊ {2,2}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {2,2}
[3]
 => []
 => []
 => ?
 => ? ∊ {3,3,3}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {3,3,3}
[1,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {3,3,3}
[4]
 => []
 => []
 => ?
 => ? ∊ {4,4,4,4}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {4,4,4,4}
[2,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[2,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {4,4,4,4}
[1,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {4,4,4,4}
[5]
 => []
 => []
 => ?
 => ? ∊ {5,5,5,5,5}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {5,5,5,5,5}
[3,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[3,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {5,5,5,5,5}
[2,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {5,5,5,5,5}
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {5,5,5,5,5}
[6]
 => []
 => []
 => ?
 => ? ∊ {6,6,6,6,6,6}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {6,6,6,6,6,6}
[4,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[4,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {6,6,6,6,6,6}
[3,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[3,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {6,6,6,6,6,6}
[2,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {6,6,6,6,6,6}
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {6,6,6,6,6,6}
[7]
 => []
 => []
 => ?
 => ? ∊ {7,7,7,7,7,7,7}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[5,2]
 => [2]
 => [1,1]
 => [1]
 => 1
[5,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[4,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[4,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
[3,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? ∊ {7,7,7,7,7,7,7}
Description
The number of monomer-dimer tilings of a Ferrers diagram. For a hook of length $n$, this is the $n$-th Fibonacci number.
The following 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000719The number of alignments in a perfect matching. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001645The pebbling number of a connected graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001626The number of maximal proper sublattices of a lattice. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St001128The exponens consonantiae of a partition. St000260The radius of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001621The number of atoms of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001926Sparre Andersen's position of the maximum of a signed permutation.