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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000946
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000946: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000946: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1,4,5,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,5,3,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,3,1,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,4,5,1,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,5,4,3,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,1,2,4,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,2,5,4] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,4,1,2,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[3,4,1,5,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,4,5,2,1] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
Description
The sum of the skew hook positions in a Dyck path.
A skew hook is an occurrence of a down step followed by two up steps or of an up step followed by a down step.
Write $U_i$ for the $i$-th up step and $D_j$ for the $j$-th down step in the Dyck path. Then the skew hook set is the set $$H = \{j: U_{i−1} U_i D_j \text{ is a skew hook}\} \cup \{i: D_{i−1} D_i U_j\text{ is a skew hook}\}.$$
This statistic is the sum of all elements in $H$.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 1 - 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 1
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,4,3,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[3,2,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,4,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[4,2,3,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[4,3,2,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,3,5,4,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,3,5,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,2,4,3] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,3,2,4] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,3,4,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,5,4,3,2] => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,4,5,3] => [2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[2,1,5,4,3] => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,1,4,5] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[2,3,1,5,4] => [3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[2,4,3,1,5] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,4,5,1,3] => [4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[2,5,3,4,1] => [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[2,5,4,3,1] => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[3,1,2,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,1,2,5,4] => [3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[3,2,1,4,5] => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,2,1,5,4] => [2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,2,4,1,5] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,2,5,4,1] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[3,4,1,2,5] => [3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,4,1,5,2] => [3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[3,4,5,2,1] => [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[3,5,1,2,4] => [3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[3,5,1,4,2] => [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,4,1,2] => [4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[4,1,3,2,5] => [3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[4,1,5,2,3] => [4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[4,2,1,3,5] => [2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[4,2,3,1,5] => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[4,2,3,5,1] => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[4,2,5,1,3] => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,2,1,5] => [3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,2,5,1] => [3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[4,3,5,1,2] => [4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[4,5,1,3,2] => [4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[4,5,2,1,3] => [4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[4,5,3,1,2] => [3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,1,3,4,2] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[5,1,4,3,2] => [4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[5,2,1,4,3] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[5,2,3,1,4] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[5,2,3,4,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[5,2,4,3,1] => [2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,3,2,1,4] => [3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[5,3,2,4,1] => [3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,3,4,2,1] => [4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[5,4,1,2,3] => [4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[5,4,2,3,1] => [4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 - 1
[5,4,3,2,1] => [3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6)
=> ? = 1 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001568
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 25%
Mp00244: Signed permutations —bar⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 25%
Values
[1,2,3] => [1,2,3] => [-1,-2,-3] => [1,1,1]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => [1,1,1,1]
=> 2 = 1 + 1
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => [1,1]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => [1,1]
=> 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => [1,1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => [1,1]
=> 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => []
=> ? = 0 + 1
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => [1,1]
=> 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => []
=> ? = 0 + 1
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => [1,1]
=> 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => []
=> ? = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => [1,1,1]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => [1,1,1]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => [3,1,1]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [-1,-2,-5,-3,-4] => [3,1,1]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => [1,1,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => [1,1,1]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => [1]
=> ? = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => [3,1,1]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => [3,1,1]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [-1,-4,-2,-3,-5] => [3,1,1]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => [1,1,1]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => [3,1,1]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => [1]
=> ? = 2 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => [3,1,1]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => [3,1,1]
=> 2 = 1 + 1
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => [1,1,1]
=> 2 = 1 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [-1,-5,-4,-3,-2] => [1]
=> ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [-2,-1,-3,-4,-5] => [1,1,1]
=> 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [-2,-1,-3,-5,-4] => [1]
=> ? = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,-1,-4,-3,-5] => [1]
=> ? = 2 + 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,-1,-4,-5,-3] => [3]
=> 1 = 0 + 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-1,-5,-3,-4] => [3]
=> 1 = 0 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [-2,-1,-5,-4,-3] => [1]
=> ? = 2 + 1
[2,3,1,4,5] => [2,3,1,4,5] => [-2,-3,-1,-4,-5] => [3,1,1]
=> 2 = 1 + 1
[2,3,1,5,4] => [2,3,1,5,4] => [-2,-3,-1,-5,-4] => [3]
=> 1 = 0 + 1
[2,4,3,1,5] => [2,4,3,1,5] => [-2,-4,-3,-1,-5] => [3,1,1]
=> 2 = 1 + 1
[2,4,5,1,3] => [2,4,5,1,3] => [-2,-4,-5,-1,-3] => [3]
=> 1 = 0 + 1
[2,5,3,4,1] => [2,5,3,4,1] => [-2,-5,-3,-4,-1] => [3,1,1]
=> 2 = 1 + 1
[2,5,4,3,1] => [2,5,4,3,1] => [-2,-5,-4,-3,-1] => [3]
=> 1 = 0 + 1
[3,1,2,4,5] => [3,1,2,4,5] => [-3,-1,-2,-4,-5] => [3,1,1]
=> 2 = 1 + 1
[3,1,2,5,4] => [3,1,2,5,4] => [-3,-1,-2,-5,-4] => [3]
=> 1 = 0 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [-3,-2,-1,-4,-5] => [1,1,1]
=> 2 = 1 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [-3,-2,-1,-5,-4] => [1]
=> ? = 2 + 1
[3,2,4,1,5] => [3,2,4,1,5] => [-3,-2,-4,-1,-5] => [3,1,1]
=> 2 = 1 + 1
[3,2,5,4,1] => [3,2,5,4,1] => [-3,-2,-5,-4,-1] => [3,1,1]
=> 2 = 1 + 1
[3,4,1,2,5] => [3,4,1,2,5] => [-3,-4,-1,-2,-5] => [1]
=> ? = 2 + 1
[3,4,1,5,2] => [3,4,1,5,2] => [-3,-4,-1,-5,-2] => [3]
=> 1 = 0 + 1
[3,4,5,2,1] => [3,4,5,2,1] => [-3,-4,-5,-2,-1] => [3]
=> 1 = 0 + 1
[3,5,1,2,4] => [3,5,1,2,4] => [-3,-5,-1,-2,-4] => [3]
=> 1 = 0 + 1
[3,5,1,4,2] => [3,5,1,4,2] => [-3,-5,-1,-4,-2] => [1]
=> ? = 2 + 1
[3,5,4,1,2] => [3,5,4,1,2] => [-3,-5,-4,-1,-2] => [3]
=> 1 = 0 + 1
[4,1,3,2,5] => [4,1,3,2,5] => [-4,-1,-3,-2,-5] => [3,1,1]
=> 2 = 1 + 1
[4,1,5,2,3] => [4,1,5,2,3] => [-4,-1,-5,-2,-3] => [3]
=> 1 = 0 + 1
[4,2,1,3,5] => [4,2,1,3,5] => [-4,-2,-1,-3,-5] => [3,1,1]
=> 2 = 1 + 1
[4,2,3,1,5] => [4,2,3,1,5] => [-4,-2,-3,-1,-5] => [1,1,1]
=> 2 = 1 + 1
[4,2,3,5,1] => [4,2,3,5,1] => [-4,-2,-3,-5,-1] => [3,1,1]
=> 2 = 1 + 1
[4,2,5,1,3] => [4,2,5,1,3] => [-4,-2,-5,-1,-3] => [1]
=> ? = 2 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [-4,-3,-2,-1,-5] => [1]
=> ? = 2 + 1
[4,3,2,5,1] => [4,3,2,5,1] => [-4,-3,-2,-5,-1] => [3]
=> 1 = 0 + 1
[4,3,5,1,2] => [4,3,5,1,2] => [-4,-3,-5,-1,-2] => [3]
=> 1 = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => [-4,-5,-1,-3,-2] => [3]
=> 1 = 0 + 1
[4,5,2,1,3] => [4,5,2,1,3] => [-4,-5,-2,-1,-3] => [3]
=> 1 = 0 + 1
[4,5,3,1,2] => [4,5,3,1,2] => [-4,-5,-3,-1,-2] => [1]
=> ? = 2 + 1
[5,2,4,3,1] => [5,2,4,3,1] => [-5,-2,-4,-3,-1] => [1]
=> ? = 2 + 1
[5,3,2,4,1] => [5,3,2,4,1] => [-5,-3,-2,-4,-1] => [1]
=> ? = 2 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [-5,-4,-3,-2,-1] => [1]
=> ? = 2 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [-1,-2,-3,-4,-5,-6] => ?
=> ? = 1 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [-1,-2,-3,-4,-6,-5] => ?
=> ? = 1 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [-1,-2,-3,-5,-4,-6] => ?
=> ? = 1 + 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => [-1,-2,-3,-5,-6,-4] => ?
=> ? = 1 + 1
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [-1,-2,-3,-6,-4,-5] => ?
=> ? = 1 + 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [-1,-2,-3,-6,-5,-4] => ?
=> ? = 1 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [-1,-2,-4,-3,-5,-6] => ?
=> ? = 1 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [-1,-2,-4,-3,-6,-5] => ?
=> ? = 2 + 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [-1,-2,-4,-5,-3,-6] => ?
=> ? = 1 + 1
[1,2,4,5,6,3] => [1,2,4,5,6,3] => [-1,-2,-4,-5,-6,-3] => ?
=> ? = 1 + 1
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [-1,-2,-4,-6,-3,-5] => ?
=> ? = 1 + 1
[1,2,4,6,5,3] => [1,2,4,6,5,3] => [-1,-2,-4,-6,-5,-3] => ?
=> ? = 1 + 1
[1,2,5,3,4,6] => [1,2,5,3,4,6] => [-1,-2,-5,-3,-4,-6] => ?
=> ? = 1 + 1
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [-1,-2,-5,-3,-6,-4] => ?
=> ? = 1 + 1
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [-1,-2,-5,-4,-3,-6] => ?
=> ? = 1 + 1
[1,2,5,4,6,3] => [1,2,5,4,6,3] => [-1,-2,-5,-4,-6,-3] => ?
=> ? = 1 + 1
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [-1,-2,-5,-6,-3,-4] => ?
=> ? = 2 + 1
[1,2,5,6,4,3] => [1,2,5,6,4,3] => [-1,-2,-5,-6,-4,-3] => ?
=> ? = 1 + 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => [-1,-2,-6,-3,-4,-5] => ?
=> ? = 1 + 1
[1,2,6,3,5,4] => [1,2,6,3,5,4] => ? => ?
=> ? = 1 + 1
[1,2,6,4,3,5] => [1,2,6,4,3,5] => ? => ?
=> ? = 1 + 1
[1,2,6,4,5,3] => [1,2,6,4,5,3] => [-1,-2,-6,-4,-5,-3] => ?
=> ? = 1 + 1
[1,2,6,5,3,4] => [1,2,6,5,3,4] => ? => ?
=> ? = 1 + 1
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [-1,-2,-6,-5,-4,-3] => ?
=> ? = 2 + 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [-1,-3,-2,-4,-5,-6] => ?
=> ? = 1 + 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [-1,-3,-2,-4,-6,-5] => ?
=> ? = 2 + 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [-1,-3,-2,-5,-4,-6] => ?
=> ? = 2 + 1
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [-1,-3,-2,-5,-6,-4] => ?
=> ? = 2 + 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [-1,-3,-2,-6,-4,-5] => ?
=> ? = 2 + 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [-1,-3,-2,-6,-5,-4] => ?
=> ? = 2 + 1
[1,3,4,2,5,6] => [1,3,4,2,5,6] => [-1,-3,-4,-2,-5,-6] => ?
=> ? = 1 + 1
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [-1,-3,-4,-2,-6,-5] => ?
=> ? = 2 + 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St001964
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 12%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 12%
Values
[1,2,3] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[1,3,2,4] => [3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 - 1
[1,4,3,2] => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 - 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,1,4,3] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 - 1
[3,2,1,4] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[3,4,1,2] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 - 1
[4,2,3,1] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[4,3,2,1] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 - 1
[1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,2,4,5,3] => [2,5,3,4,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,2,5,3,4] => [2,4,5,3,1] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,2,5,4,3] => [2,5,4,3,1] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,3,2,4,5] => [3,2,4,5,1] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,3,2,5,4] => [3,2,5,4,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[1,3,4,2,5] => [4,2,3,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 1 - 1
[1,3,5,4,2] => [5,2,4,3,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,4,2,3,5] => [3,4,2,5,1] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,4,3,2,5] => [4,3,2,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 1 - 1
[1,4,3,5,2] => [5,3,2,4,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,4,5,2,3] => [4,5,2,3,1] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 - 1
[1,5,2,4,3] => [3,5,4,2,1] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 - 1
[1,5,3,2,4] => [4,3,5,2,1] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 1 - 1
[1,5,3,4,2] => [5,3,4,2,1] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[1,5,4,3,2] => [5,4,3,2,1] => [1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 2 - 1
[2,1,3,4,5] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,1,3,5,4] => [1,3,5,4,2] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[2,1,4,3,5] => [1,4,3,5,2] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[2,1,4,5,3] => [1,5,3,4,2] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
[2,1,5,3,4] => [1,4,5,3,2] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
[2,1,5,4,3] => [1,5,4,3,2] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[2,3,1,4,5] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,3,1,5,4] => [1,2,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 - 1
[2,4,3,1,5] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,4,5,1,3] => [1,5,2,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 - 1
[2,5,3,4,1] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,5,4,3,1] => [1,4,3,2,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
[3,1,2,4,5] => [3,1,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[3,1,2,5,4] => [3,1,5,4,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 - 1
[3,2,1,4,5] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,2,1,5,4] => [2,1,5,4,3] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 - 1
[3,2,4,1,5] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,2,5,4,1] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[3,4,1,2,5] => [4,1,2,5,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 - 1
[3,4,1,5,2] => [5,1,2,4,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 - 1
[3,4,5,2,1] => [4,1,2,3,5] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 - 1
[3,5,1,2,4] => [4,1,5,2,3] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 - 1
[3,5,1,4,2] => [5,1,4,2,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[3,5,4,1,2] => [5,1,3,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 - 1
[4,1,3,2,5] => [4,3,1,5,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 1 - 1
[4,1,5,2,3] => [4,5,1,3,2] => [1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 - 1
[4,2,1,3,5] => [2,4,1,5,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 1 - 1
[4,2,3,1,5] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,2,3,5,1] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[4,2,5,1,3] => [2,5,1,3,4] => [1,2,5,4,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 - 1
[4,3,2,1,5] => [3,2,1,5,4] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[4,3,2,5,1] => [3,2,1,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
[4,3,5,1,2] => [5,2,1,3,4] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 - 1
[4,5,1,3,2] => [5,4,1,2,3] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 - 1
[4,5,2,1,3] => [3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 - 1
[4,5,3,1,2] => [5,3,1,2,4] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 - 1
[5,2,3,4,1] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,1,3,4,5,6] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,1,4,5,6] => [1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,5,4,1,6] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,3,6,4,5,1] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,3,1,5,6] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,3,5,1,6] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,4,3,6,5,1] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,5,3,4,1,6] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,5,3,4,6,1] => [1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[2,6,3,4,5,1] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[3,2,1,4,5,6] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[3,2,4,1,5,6] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[3,2,4,5,1,6] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[3,2,4,6,5,1] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[3,2,5,4,1,6] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[3,2,5,4,6,1] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[3,2,6,4,5,1] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[4,2,3,1,5,6] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[4,2,3,5,1,6] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[4,2,3,5,6,1] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[4,2,3,6,5,1] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[5,2,3,4,1,6] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[5,2,3,4,6,1] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[6,2,3,4,5,1] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
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