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Your data matches 73 different statistics following compositions of up to 3 maps.
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Matching statistic: St001086
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [(1,2)]
=> [2,1] => 0
[1,2] => [[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [3,4,2,1] => 0
[2,1] => [[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => 0
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 2
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 3
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 3
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => 3
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => 0
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[3,5,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[4,5,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => 4
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[2,1,5,3,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,1,4,2,6,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,1,5,2,6,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,1,5,6,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,2,5,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,2,5,6,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,4,1,2,6,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,5,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,5,1,6,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,5,2,4,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,5,2,6,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,5,2,6,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,6,1,2,5,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,6,1,5,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,6,2,5,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[3,6,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[4,5,1,6,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[4,5,2,3,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[4,5,2,6,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[4,6,1,5,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
[4,6,2,5,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => 5
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St001135
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [(1,2)]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,5,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,5,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[2,1,5,3,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,1,4,2,6,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,1,5,2,6,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,1,5,6,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,2,5,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,2,5,6,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,4,1,2,6,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,1,6,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,2,4,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,2,6,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,2,6,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,1,2,5,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,1,5,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,2,5,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,1,6,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,2,3,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,2,6,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,6,1,5,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,6,2,5,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001201
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [(1,2)]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,5,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,5,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[2,1,5,3,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,1,4,2,6,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,1,5,2,6,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,1,5,6,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,2,5,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,2,5,6,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,4,1,2,6,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,1,6,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,2,4,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,2,6,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,2,6,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,1,2,5,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,1,5,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,2,5,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,1,6,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,2,3,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,2,6,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,6,1,5,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,6,2,5,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
Description
The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path.
Matching statistic: St000678
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [(1,2)]
=> [1,0]
=> ? = 0 + 1
[1,2] => [[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,5,3,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1,5,4,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,5,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,5,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,5,2,3,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[2,1,5,3,6,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,1,4,2,6,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,1,5,2,6,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,1,5,6,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,2,5,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,2,5,6,1,4] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,2,5,6,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,4,1,2,6,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,1,6,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,2,4,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,2,6,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,5,2,6,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,1,2,5,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,1,5,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,2,5,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[3,6,2,5,4,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,1,6,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,2,3,6,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,2,6,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,5,2,6,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,6,1,5,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,6,2,5,1,3] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,6,2,5,3,1] => [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001603
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 80%●distinct values known / distinct values provided: 33%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 80%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1]
=> []
=> ? = 0 - 3
[1,2] => [1,2] => [1,1]
=> [1]
=> ? = 0 - 3
[2,1] => [2,1] => [2]
=> []
=> ? = 1 - 3
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> ? = 0 - 3
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 3
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? = 2 - 3
[2,3,1] => [3,2,1] => [2,1]
=> [1]
=> ? = 1 - 3
[3,1,2] => [3,2,1] => [2,1]
=> [1]
=> ? = 2 - 3
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? = 1 - 3
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> ? = 3 - 3
[3,1,4,2] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? = 3 - 3
[3,4,1,2] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 3 - 3
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> ? = 0 - 3
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,1,4,2,5] => [4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 4 - 3
[3,1,5,2,4] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,1,5,4,2] => [5,2,4,3,1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,2,5,4,1] => [5,2,4,3,1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,4,1,2,5] => [4,3,2,1,5] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,5,1,2,4] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,5,1,4,2] => [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[3,5,2,4,1] => [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[4,5,1,3,2] => [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[4,5,2,3,1] => [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 1 = 4 - 3
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,2,2]
=> [2,2]
=> 2 = 5 - 3
[2,1,5,3,6,4] => [2,1,6,4,5,3] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[2,1,5,6,3,4] => [2,1,6,5,4,3] => [2,2,2]
=> [2,2]
=> 2 = 5 - 3
[3,1,4,2,6,5] => [4,2,3,1,6,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,1,5,2,6,4] => [4,2,6,1,5,3] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,1,5,6,2,4] => [5,2,6,4,1,3] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,2,5,6,1,4] => [5,2,6,4,1,3] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,2,5,6,4,1] => [6,2,5,4,3,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,4,1,2,6,5] => [4,3,2,1,6,5] => [2,2,2]
=> [2,2]
=> 2 = 5 - 3
[3,5,1,2,6,4] => [4,6,3,1,5,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,5,1,6,2,4] => [5,6,3,4,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,5,2,4,6,1] => [6,4,3,2,5,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,5,2,6,1,4] => [5,6,3,4,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,5,2,6,4,1] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,6,1,2,5,4] => [4,6,3,1,5,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,6,1,5,2,4] => [5,6,3,4,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,6,2,5,1,4] => [5,6,3,4,1,2] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[3,6,2,5,4,1] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[4,5,1,6,2,3] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[4,5,2,3,6,1] => [6,4,3,2,5,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[4,5,2,6,1,3] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[4,5,2,6,3,1] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[4,6,1,5,2,3] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[4,6,2,5,1,3] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 2 = 5 - 3
[4,6,2,5,3,1] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 2 = 5 - 3
[4,6,5,2,1,3] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 2 = 5 - 3
[4,6,5,2,3,1] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 2 = 5 - 3
[6,1,3,2,5,4] => [6,2,4,3,5,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[6,1,3,5,2,4] => [6,2,5,4,3,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[6,1,4,5,2,3] => [6,2,5,4,3,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[6,3,1,2,5,4] => [6,4,3,2,5,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[6,3,1,5,2,4] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[6,3,2,5,1,4] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[6,3,2,5,4,1] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[6,4,1,5,2,3] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 2 = 5 - 3
[6,4,2,5,1,3] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 2 = 5 - 3
[6,4,2,5,3,1] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 2 = 5 - 3
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000264
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 55%●distinct values known / distinct values provided: 17%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 55%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,2] => [1,2] => [2] => ([],2)
=> ? = 0 - 2
[2,1] => [2,1] => [2] => ([],2)
=> ? = 1 - 2
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 0 - 2
[1,3,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? = 1 - 2
[2,1,3] => [2,1,3] => [3] => ([],3)
=> ? = 2 - 2
[2,3,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 - 2
[3,1,2] => [3,1,2] => [3] => ([],3)
=> ? = 2 - 2
[3,2,1] => [2,3,1] => [3] => ([],3)
=> ? = 1 - 2
[2,1,4,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 3 - 2
[3,1,4,2] => [4,3,1,2] => [1,3] => ([(2,3)],4)
=> ? = 3 - 2
[3,4,1,2] => [4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3 - 2
[4,3,2,1] => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 - 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[2,1,5,3,4] => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 4 - 2
[3,1,4,2,5] => [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> ? = 4 - 2
[3,1,5,2,4] => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? = 4 - 2
[3,1,5,4,2] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,2,5,4,1] => [2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,4,1,2,5] => [4,1,3,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,5,1,2,4] => [5,1,3,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,5,1,4,2] => [4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[3,5,2,4,1] => [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[4,5,1,3,2] => [5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[4,5,2,3,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 4 - 2
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[2,1,5,3,6,4] => [2,1,6,5,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[2,1,5,6,3,4] => [2,1,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,1,4,2,6,5] => [4,3,1,2,6,5] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,1,5,2,6,4] => [6,5,3,1,2,4] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,1,5,6,2,4] => [6,3,1,2,5,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,2,5,4,6,1] => [2,4,6,5,3,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,2,5,6,1,4] => [2,6,3,1,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,2,5,6,4,1] => [2,6,4,5,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,4,1,2,6,5] => [4,1,3,2,6,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,5,1,2,6,4] => [6,5,1,3,2,4] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,5,1,6,2,4] => [6,1,3,2,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,5,2,4,6,1] => [4,6,5,2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,5,2,6,1,4] => [6,2,3,1,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,5,2,6,4,1] => [6,4,5,2,3,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,6,1,2,5,4] => [5,6,1,3,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 2
[3,6,1,5,2,4] => [5,1,3,2,6,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,6,2,5,1,4] => [5,2,3,1,6,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[3,6,2,5,4,1] => [5,4,6,2,3,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,5,1,6,2,3] => [6,1,5,4,2,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,5,2,3,6,1] => [6,5,2,3,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,5,2,6,1,3] => [6,2,5,4,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,5,2,6,3,1] => [6,2,3,5,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,6,1,5,2,3] => [5,1,6,4,2,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,6,2,5,1,3] => [5,2,6,4,1,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,6,2,5,3,1] => [5,2,3,6,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,6,5,2,1,3] => [5,2,6,1,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[4,6,5,2,3,1] => [5,2,6,3,4,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[6,1,3,2,5,4] => [3,5,6,1,2,4] => [6] => ([],6)
=> ? = 5 - 2
[6,1,3,5,2,4] => [3,5,1,2,6,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 5 - 2
[6,1,4,5,2,3] => [5,4,1,2,6,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[6,3,1,2,5,4] => [3,1,5,6,2,4] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 5 - 2
[6,3,1,5,2,4] => [3,1,5,2,6,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[6,3,2,5,1,4] => [3,2,5,1,6,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[6,3,2,5,4,1] => [3,2,5,4,6,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[6,4,1,5,2,3] => [5,4,1,6,2,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[6,4,2,5,1,3] => [5,4,2,6,1,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[6,4,2,5,3,1] => [5,4,2,3,6,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[6,4,5,2,1,3] => [5,2,4,1,6,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
[6,4,5,2,3,1] => [5,2,4,3,6,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 5 - 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001629
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 17% ●values known / values provided: 55%●distinct values known / distinct values provided: 17%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 17% ●values known / values provided: 55%●distinct values known / distinct values provided: 17%
Values
[1] => [1] => [1] => [1] => ? = 0 - 4
[1,2] => [1,2] => [2] => [1] => ? = 0 - 4
[2,1] => [2,1] => [2] => [1] => ? = 1 - 4
[1,2,3] => [1,2,3] => [3] => [1] => ? = 0 - 4
[1,3,2] => [1,3,2] => [1,2] => [1,1] => ? = 1 - 4
[2,1,3] => [2,1,3] => [3] => [1] => ? = 2 - 4
[2,3,1] => [3,2,1] => [2,1] => [1,1] => ? = 1 - 4
[3,1,2] => [3,1,2] => [3] => [1] => ? = 2 - 4
[3,2,1] => [2,3,1] => [3] => [1] => ? = 1 - 4
[2,1,4,3] => [2,1,4,3] => [2,2] => [2] => ? = 3 - 4
[3,1,4,2] => [4,3,1,2] => [1,3] => [1,1] => ? = 3 - 4
[3,4,1,2] => [4,1,3,2] => [3,1] => [1,1] => ? = 3 - 4
[4,3,2,1] => [3,2,4,1] => [2,2] => [2] => ? = 0 - 4
[2,1,4,3,5] => [2,1,4,3,5] => [2,3] => [1,1] => ? = 4 - 4
[2,1,5,3,4] => [2,1,5,3,4] => [2,3] => [1,1] => ? = 4 - 4
[3,1,4,2,5] => [4,3,1,2,5] => [1,4] => [1,1] => ? = 4 - 4
[3,1,5,2,4] => [5,3,1,2,4] => [1,4] => [1,1] => ? = 4 - 4
[3,1,5,4,2] => [4,5,3,1,2] => [3,2] => [1,1] => ? = 4 - 4
[3,2,5,4,1] => [2,4,5,3,1] => [4,1] => [1,1] => ? = 4 - 4
[3,4,1,2,5] => [4,1,3,2,5] => [3,2] => [1,1] => ? = 4 - 4
[3,5,1,2,4] => [5,1,3,2,4] => [3,2] => [1,1] => ? = 4 - 4
[3,5,1,4,2] => [4,5,1,3,2] => [4,1] => [1,1] => ? = 4 - 4
[3,5,2,4,1] => [4,5,2,3,1] => [4,1] => [1,1] => ? = 4 - 4
[4,5,1,3,2] => [5,1,3,4,2] => [4,1] => [1,1] => ? = 4 - 4
[4,5,2,3,1] => [5,2,3,4,1] => [4,1] => [1,1] => ? = 4 - 4
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [2,2,2] => [3] => 1 = 5 - 4
[2,1,5,3,6,4] => [2,1,6,5,3,4] => [2,1,3] => [1,1,1] => 1 = 5 - 4
[2,1,5,6,3,4] => [2,1,6,3,5,4] => [2,3,1] => [1,1,1] => 1 = 5 - 4
[3,1,4,2,6,5] => [4,3,1,2,6,5] => [1,3,2] => [1,1,1] => 1 = 5 - 4
[3,1,5,2,6,4] => [6,5,3,1,2,4] => [1,2,3] => [1,1,1] => 1 = 5 - 4
[3,1,5,6,2,4] => [6,3,1,2,5,4] => [1,4,1] => [1,1,1] => 1 = 5 - 4
[3,2,5,4,6,1] => [2,4,6,5,3,1] => [3,2,1] => [1,1,1] => 1 = 5 - 4
[3,2,5,6,1,4] => [2,6,3,1,5,4] => [3,2,1] => [1,1,1] => 1 = 5 - 4
[3,2,5,6,4,1] => [2,6,4,5,3,1] => [2,3,1] => [1,1,1] => 1 = 5 - 4
[3,4,1,2,6,5] => [4,1,3,2,6,5] => [3,1,2] => [1,1,1] => 1 = 5 - 4
[3,5,1,2,6,4] => [6,5,1,3,2,4] => [1,3,2] => [1,1,1] => 1 = 5 - 4
[3,5,1,6,2,4] => [6,1,3,2,5,4] => [3,2,1] => [1,1,1] => 1 = 5 - 4
[3,5,2,4,6,1] => [4,6,5,2,3,1] => [2,3,1] => [1,1,1] => 1 = 5 - 4
[3,5,2,6,1,4] => [6,2,3,1,5,4] => [3,2,1] => [1,1,1] => 1 = 5 - 4
[3,5,2,6,4,1] => [6,4,5,2,3,1] => [1,4,1] => [1,1,1] => 1 = 5 - 4
[3,6,1,2,5,4] => [5,6,1,3,2,4] => [4,2] => [1,1] => ? = 5 - 4
[3,6,1,5,2,4] => [5,1,3,2,6,4] => [3,1,2] => [1,1,1] => 1 = 5 - 4
[3,6,2,5,1,4] => [5,2,3,1,6,4] => [3,1,2] => [1,1,1] => 1 = 5 - 4
[3,6,2,5,4,1] => [5,4,6,2,3,1] => [1,4,1] => [1,1,1] => 1 = 5 - 4
[4,5,1,6,2,3] => [6,1,5,4,2,3] => [2,2,2] => [3] => 1 = 5 - 4
[4,5,2,3,6,1] => [6,5,2,3,4,1] => [1,4,1] => [1,1,1] => 1 = 5 - 4
[4,5,2,6,1,3] => [6,2,5,4,1,3] => [2,2,2] => [3] => 1 = 5 - 4
[4,5,2,6,3,1] => [6,2,3,5,4,1] => [3,2,1] => [1,1,1] => 1 = 5 - 4
[4,6,1,5,2,3] => [5,1,6,4,2,3] => [2,2,2] => [3] => 1 = 5 - 4
[4,6,2,5,1,3] => [5,2,6,4,1,3] => [2,2,2] => [3] => 1 = 5 - 4
[4,6,2,5,3,1] => [5,2,3,6,4,1] => [3,2,1] => [1,1,1] => 1 = 5 - 4
[4,6,5,2,1,3] => [5,2,6,1,4,3] => [2,3,1] => [1,1,1] => 1 = 5 - 4
[4,6,5,2,3,1] => [5,2,6,3,4,1] => [2,3,1] => [1,1,1] => 1 = 5 - 4
[6,1,3,2,5,4] => [3,5,6,1,2,4] => [6] => [1] => ? = 5 - 4
[6,1,3,5,2,4] => [3,5,1,2,6,4] => [4,2] => [1,1] => ? = 5 - 4
[6,1,4,5,2,3] => [5,4,1,2,6,3] => [1,3,2] => [1,1,1] => 1 = 5 - 4
[6,3,1,2,5,4] => [3,1,5,6,2,4] => [2,4] => [1,1] => ? = 5 - 4
[6,3,1,5,2,4] => [3,1,5,2,6,4] => [2,2,2] => [3] => 1 = 5 - 4
[6,3,2,5,1,4] => [3,2,5,1,6,4] => [2,2,2] => [3] => 1 = 5 - 4
[6,3,2,5,4,1] => [3,2,5,4,6,1] => [2,2,2] => [3] => 1 = 5 - 4
[6,4,1,5,2,3] => [5,4,1,6,2,3] => [1,2,3] => [1,1,1] => 1 = 5 - 4
[6,4,2,5,1,3] => [5,4,2,6,1,3] => [1,2,3] => [1,1,1] => 1 = 5 - 4
[6,4,2,5,3,1] => [5,4,2,3,6,1] => [1,3,2] => [1,1,1] => 1 = 5 - 4
[6,4,5,2,1,3] => [5,2,4,1,6,3] => [2,2,2] => [3] => 1 = 5 - 4
[6,4,5,2,3,1] => [5,2,4,3,6,1] => [2,2,2] => [3] => 1 = 5 - 4
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001605
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 32%●distinct values known / distinct values provided: 17%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 32%●distinct values known / distinct values provided: 17%
Values
[1] => [] => []
=> ?
=> ? = 0 - 4
[1,2] => [1] => [1]
=> []
=> ? = 0 - 4
[2,1] => [1] => [1]
=> []
=> ? = 1 - 4
[1,2,3] => [1,2] => [2]
=> []
=> ? = 0 - 4
[1,3,2] => [1,2] => [2]
=> []
=> ? = 1 - 4
[2,1,3] => [2,1] => [1,1]
=> [1]
=> ? = 2 - 4
[2,3,1] => [2,1] => [1,1]
=> [1]
=> ? = 1 - 4
[3,1,2] => [1,2] => [2]
=> []
=> ? = 2 - 4
[3,2,1] => [2,1] => [1,1]
=> [1]
=> ? = 1 - 4
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? = 3 - 4
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> ? = 3 - 4
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? = 3 - 4
[4,3,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> ? = 0 - 4
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> ? = 4 - 4
[2,1,5,3,4] => [2,1,3,4] => [3,1]
=> [1]
=> ? = 4 - 4
[3,1,4,2,5] => [3,1,4,2] => [2,2]
=> [2]
=> ? = 4 - 4
[3,1,5,2,4] => [3,1,2,4] => [3,1]
=> [1]
=> ? = 4 - 4
[3,1,5,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> ? = 4 - 4
[3,2,5,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> ? = 4 - 4
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> ? = 4 - 4
[3,5,1,2,4] => [3,1,2,4] => [3,1]
=> [1]
=> ? = 4 - 4
[3,5,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> ? = 4 - 4
[3,5,2,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> ? = 4 - 4
[4,5,1,3,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> ? = 4 - 4
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? = 4 - 4
[2,1,4,3,6,5] => [2,1,4,3,5] => [3,2]
=> [2]
=> ? = 5 - 4
[2,1,5,3,6,4] => [2,1,5,3,4] => [3,2]
=> [2]
=> ? = 5 - 4
[2,1,5,6,3,4] => [2,1,5,3,4] => [3,2]
=> [2]
=> ? = 5 - 4
[3,1,4,2,6,5] => [3,1,4,2,5] => [3,2]
=> [2]
=> ? = 5 - 4
[3,1,5,2,6,4] => [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 4
[3,1,5,6,2,4] => [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 4
[3,2,5,4,6,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,5,6,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,5,6,4,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,4,1,2,6,5] => [3,4,1,2,5] => [3,2]
=> [2]
=> ? = 5 - 4
[3,5,1,2,6,4] => [3,5,1,2,4] => [3,2]
=> [2]
=> ? = 5 - 4
[3,5,1,6,2,4] => [3,5,1,2,4] => [3,2]
=> [2]
=> ? = 5 - 4
[3,5,2,4,6,1] => [3,5,2,4,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,5,2,6,1,4] => [3,5,2,1,4] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,5,2,6,4,1] => [3,5,2,4,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,6,1,2,5,4] => [3,1,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 4
[3,6,1,5,2,4] => [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 4
[3,6,2,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,6,2,5,4,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,5,1,6,2,3] => [4,5,1,2,3] => [3,2]
=> [2]
=> ? = 5 - 4
[4,5,2,3,6,1] => [4,5,2,3,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,5,2,6,1,3] => [4,5,2,1,3] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,5,2,6,3,1] => [4,5,2,3,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,6,1,5,2,3] => [4,1,5,2,3] => [3,2]
=> [2]
=> ? = 5 - 4
[4,6,2,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,6,2,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,6,5,2,1,3] => [4,5,2,1,3] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,6,5,2,3,1] => [4,5,2,3,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[6,1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 4
[6,1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 4
[6,1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 5 - 4
[6,3,1,2,5,4] => [3,1,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 4
[6,3,1,5,2,4] => [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 4
[6,3,2,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[6,3,2,5,4,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[6,4,1,5,2,3] => [4,1,5,2,3] => [3,2]
=> [2]
=> ? = 5 - 4
[6,4,2,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[6,4,2,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[6,4,5,2,1,3] => [4,5,2,1,3] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
[6,4,5,2,3,1] => [4,5,2,3,1] => [2,2,1]
=> [2,1]
=> 1 = 5 - 4
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001604
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 32%●distinct values known / distinct values provided: 17%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 32%●distinct values known / distinct values provided: 17%
Values
[1] => [] => []
=> ?
=> ? = 0 - 5
[1,2] => [1] => [1]
=> []
=> ? = 0 - 5
[2,1] => [1] => [1]
=> []
=> ? = 1 - 5
[1,2,3] => [1,2] => [2]
=> []
=> ? = 0 - 5
[1,3,2] => [1,2] => [2]
=> []
=> ? = 1 - 5
[2,1,3] => [2,1] => [1,1]
=> [1]
=> ? = 2 - 5
[2,3,1] => [2,1] => [1,1]
=> [1]
=> ? = 1 - 5
[3,1,2] => [1,2] => [2]
=> []
=> ? = 2 - 5
[3,2,1] => [2,1] => [1,1]
=> [1]
=> ? = 1 - 5
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? = 3 - 5
[3,1,4,2] => [3,1,2] => [2,1]
=> [1]
=> ? = 3 - 5
[3,4,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? = 3 - 5
[4,3,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> ? = 0 - 5
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> ? = 4 - 5
[2,1,5,3,4] => [2,1,3,4] => [3,1]
=> [1]
=> ? = 4 - 5
[3,1,4,2,5] => [3,1,4,2] => [2,2]
=> [2]
=> ? = 4 - 5
[3,1,5,2,4] => [3,1,2,4] => [3,1]
=> [1]
=> ? = 4 - 5
[3,1,5,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> ? = 4 - 5
[3,2,5,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> ? = 4 - 5
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> ? = 4 - 5
[3,5,1,2,4] => [3,1,2,4] => [3,1]
=> [1]
=> ? = 4 - 5
[3,5,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> ? = 4 - 5
[3,5,2,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> ? = 4 - 5
[4,5,1,3,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> ? = 4 - 5
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? = 4 - 5
[2,1,4,3,6,5] => [2,1,4,3,5] => [3,2]
=> [2]
=> ? = 5 - 5
[2,1,5,3,6,4] => [2,1,5,3,4] => [3,2]
=> [2]
=> ? = 5 - 5
[2,1,5,6,3,4] => [2,1,5,3,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,1,4,2,6,5] => [3,1,4,2,5] => [3,2]
=> [2]
=> ? = 5 - 5
[3,1,5,2,6,4] => [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,1,5,6,2,4] => [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,2,5,4,6,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[3,2,5,6,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[3,2,5,6,4,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[3,4,1,2,6,5] => [3,4,1,2,5] => [3,2]
=> [2]
=> ? = 5 - 5
[3,5,1,2,6,4] => [3,5,1,2,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,5,1,6,2,4] => [3,5,1,2,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,5,2,4,6,1] => [3,5,2,4,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[3,5,2,6,1,4] => [3,5,2,1,4] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[3,5,2,6,4,1] => [3,5,2,4,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[3,6,1,2,5,4] => [3,1,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,6,1,5,2,4] => [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 5
[3,6,2,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[3,6,2,5,4,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[4,5,1,6,2,3] => [4,5,1,2,3] => [3,2]
=> [2]
=> ? = 5 - 5
[4,5,2,3,6,1] => [4,5,2,3,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[4,5,2,6,1,3] => [4,5,2,1,3] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[4,5,2,6,3,1] => [4,5,2,3,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[4,6,1,5,2,3] => [4,1,5,2,3] => [3,2]
=> [2]
=> ? = 5 - 5
[4,6,2,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[4,6,2,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[4,6,5,2,1,3] => [4,5,2,1,3] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[4,6,5,2,3,1] => [4,5,2,3,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[6,1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 5
[6,1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 5
[6,1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 5 - 5
[6,3,1,2,5,4] => [3,1,2,5,4] => [3,2]
=> [2]
=> ? = 5 - 5
[6,3,1,5,2,4] => [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 5 - 5
[6,3,2,5,1,4] => [3,2,5,1,4] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[6,3,2,5,4,1] => [3,2,5,4,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[6,4,1,5,2,3] => [4,1,5,2,3] => [3,2]
=> [2]
=> ? = 5 - 5
[6,4,2,5,1,3] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[6,4,2,5,3,1] => [4,2,5,3,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[6,4,5,2,1,3] => [4,5,2,1,3] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
[6,4,5,2,3,1] => [4,5,2,3,1] => [2,2,1]
=> [2,1]
=> 0 = 5 - 5
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001232
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,1,4,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,4,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? = 4
[3,1,4,2,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[3,1,5,2,4] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? = 4
[3,1,5,4,2] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? = 4
[3,2,5,4,1] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? = 4
[3,4,1,2,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? = 4
[3,5,1,2,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 4
[3,5,1,4,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 4
[3,5,2,4,1] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 4
[4,5,1,3,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 4
[4,5,2,3,1] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 4
[2,1,4,3,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[2,1,5,3,6,4] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[2,1,5,6,3,4] => [2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 5
[3,1,4,2,6,5] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[3,1,5,2,6,4] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[3,1,5,6,2,4] => [2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 5
[3,2,5,4,6,1] => [2,1,4,3,6,5] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[3,2,5,6,1,4] => [2,1,4,6,3,5] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 5
[3,2,5,6,4,1] => [2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 5
[3,4,1,2,6,5] => [2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 5
[3,5,1,2,6,4] => [2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 5
[3,5,1,6,2,4] => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 5
[3,5,2,4,6,1] => [2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 5
[3,5,2,6,1,4] => [1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 5
[3,5,2,6,4,1] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[3,6,1,2,5,4] => [3,6,1,2,5,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[3,6,1,5,2,4] => [3,6,1,5,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[3,6,2,5,1,4] => [3,6,2,5,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[3,6,2,5,4,1] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[4,5,1,6,2,3] => [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 5
[4,5,2,3,6,1] => [2,4,1,3,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 5
[4,5,2,6,1,3] => [1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 5
[4,5,2,6,3,1] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 5
[4,6,1,5,2,3] => [3,6,1,5,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[4,6,2,5,1,3] => [3,6,2,5,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[4,6,2,5,3,1] => [2,6,1,5,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[4,6,5,2,1,3] => [3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 5
[4,6,5,2,3,1] => [2,6,5,1,4,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 5
[6,1,3,2,5,4] => [6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,1,3,5,2,4] => [6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,1,4,5,2,3] => [6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,3,1,2,5,4] => [6,3,1,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,3,1,5,2,4] => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,3,2,5,1,4] => [6,3,2,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,3,2,5,4,1] => [6,2,1,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,4,1,5,2,3] => [6,3,1,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,4,2,5,1,3] => [6,3,2,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,4,2,5,3,1] => [6,2,1,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,4,5,2,1,3] => [6,3,5,2,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[6,4,5,2,3,1] => [6,2,5,1,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St001469The holeyness of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001061The number of indices that are both descents and recoils of a permutation. St001737The number of descents of type 2 in a permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000477The weight of a partition according to Alladi. 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. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.
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