Your data matches 6 different statistics following compositions of up to 3 maps.
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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
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => [1,2,3,6,4,5] => 2
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => [1,5,2,3,4] => 3
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5,3,4,6,2] => [1,2,5,6,3,4] => 2
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [1,2,3,6,4,5] => 2
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => 1
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
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => 0 = 1 - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,6,4,5,3] => [1,2,3,6,4,5] => 1 = 2 - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => 0 = 1 - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5,2,3,4,1] => [1,5,2,3,4] => 2 = 3 - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,5,3,4,6,2] => [1,2,5,6,3,4] => 1 = 2 - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => [1,2,3,6,4,5] => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => 0 = 1 - 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: St000460
(load all 3 compositions to match this statistic)
Mapping
St000460: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●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
[6,2]
 => 1
[5,3]
 => 2
[5,2,1]
 => 1
[4,4]
 => 3
[4,3,1]
 => 2
[4,2,2]
 => 2
[4,2,1,1]
 => 1
[5,4,4]
 => ? = 2
[4,3,3,3]
 => ? = 2
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St001880
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: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 71%
Values
[2]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2
[1,1]
 => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],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
[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
[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
[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)
 => ? = 2
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 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)
 => 6
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 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
[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
[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)
 => ? = 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
[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)
 => ? = 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 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
[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)
 => ? = 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
[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
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ? = 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 2
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ? = 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? = 2
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 2
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [.,[.,[[.,.],[[[.,.],.],.]]]]
 => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => ? = 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? = 2
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [.,[[.,.],[[[[.,.],.],.],.]]]
 => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
 => ? = 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [[.,.],[[[[[.,.],.],.],.],.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ? = 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ? = 2
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 3
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => ? = 2
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[.,[[[.,.],.],[.,.]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ? = 2
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[.,[.,[.,.]]],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 3
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [.,[[.,[.,.]],[[[.,.],.],.]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => ? = 2
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => ? = 2
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[.,[[.,.],.]],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [[.,[.,.]],[[[[.,.],.],.],.]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ? = 2
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 3
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [.,[[[.,.],.],[[[.,.],.],.]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => ? = 2
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [[[.,.],.],[[[[.,.],.],.],.]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ? = 2
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => ? = 3
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4
[5,4,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [.,[[.,[.,[.,.]]],[[.,.],.]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => ? = 3
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: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 71%
Values
[2]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2 - 1
[1,1]
 => [1,1,0,0]
 => [[.,.],.]
 => ([(0,1)],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
[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
[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
[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)
 => ? = 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
[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)
 => ? = 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
[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
[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)
 => ? = 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
[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)
 => ? = 2 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 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
[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)
 => ? = 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
[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
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[.,[.,[[.,.],[.,.]]]]]]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ? = 1 - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 2 - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ? = 1 - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? = 2 - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 2 - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [.,[.,[[.,.],[[[.,.],.],.]]]]
 => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => ? = 1 - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? = 2 - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [.,[[.,.],[[[[.,.],.],.],.]]]
 => ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
 => ? = 1 - 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 - 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 - 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [[.,.],[[[[[.,.],.],.],.],.]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ? = 1 - 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [.,[.,[.,[[.,[.,.]],[.,.]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ? = 2 - 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 3 - 1
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => ? = 2 - 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[.,[[[.,.],.],[.,.]]]]]
 => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
 => ? = 2 - 1
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [[.,[.,[.,.]]],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 - 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 3 - 1
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [.,[[.,[.,.]],[[[.,.],.],.]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => ? = 2 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
 => ? = 2 - 1
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 - 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[.,[[.,.],.]],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 - 1
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [[.,[.,.]],[[[[.,.],.],.],.]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ? = 2 - 1
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 3 - 1
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [.,[[[.,.],.],[[[.,.],.],.]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => ? = 2 - 1
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [[[[.,.],.],.],[[.,.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 - 1
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [[[.,.],.],[[[[.,.],.],.],.]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ? = 2 - 1
[6,4]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [.,[.,[[.,[.,[.,.]]],[.,.]]]]
 => ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
 => ? = 3 - 1
[5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4 - 1
[5,4,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [.,[[.,[.,[.,.]]],[[.,.],.]]]
 => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
 => ? = 3 - 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: St000718
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 71%
Values
[2]
 => [1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1]
 => [1,1,0,0]
 => [[[]]]
 => ([(0,2),(1,2)],3)
 => 3 = 2 + 1
[3]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 3 + 1
[2,1]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 + 1
[2,2]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 4 + 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 5 + 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 5 + 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 5 + 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 6 + 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 1 + 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[],[],[],[[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 6 + 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 2 + 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 1 + 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[],[],[[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 6 + 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[[[]],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 1 + 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[],[[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 6 + 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 7 + 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 7 + 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 1 + 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [[],[],[],[],[[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 7 + 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2 + 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [[],[],[[[]],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 1 + 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [[],[],[],[[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ? = 7 + 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2 + 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 + 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [[],[[[]],[],[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 1 + 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [[],[],[[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ? = 7 + 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[[[[]]],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 2 + 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[[[]],[],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [[],[[],[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 7 + 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [[[],[],[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 7 + 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[],[],[],[],[[[]]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? = 1 + 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [[],[],[[[],[]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ? = 2 + 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [[],[],[],[[[]],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
 => ? = 1 + 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 3 + 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [[],[[[],[]],[]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 2 + 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[],[],[[[[]]]]]
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ? = 2 + 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [[],[],[[[]],[],[]]]
 => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 1 + 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[[[],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[[[],[]],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 2 + 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [[],[[[[]]],[]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? = 2 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [[],[[[]],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 1 + 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[[[[],[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 3 + 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [[[[[]]],[],[]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 2 + 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [[[[]],[],[],[],[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? = 1 + 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [[],[],[],[[[],[]]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
 => ? = 2 + 1
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [[],[[[],[],[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 3 + 1
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [[],[],[[[],[]],[]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 2 + 1
Description
The largest Laplacian eigenvalue of a graph if it is integral. This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral. Various results are collected in Section 3.9 of [1]