Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001043
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [(1,4),(2,3)]
=> 1
1 => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> 1
00 => [3] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
01 => [2,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 2
10 => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 2
000 => [4] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 2
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> 2
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 1
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> 2
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> 1
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> 2
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> 2
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 2
=> [1] => [1,0]
=> [(1,2)]
=> 1
Description
The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Mp00278: Binary words rowmotionBinary words
Mp00097: Binary words delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000264: Graphs ⟶ ℤResult quality: 31% values known / values provided: 31%distinct values known / distinct values provided: 50%
Values
0 => 0 => [1] => ([],1)
=> ? = 1 + 1
1 => 1 => [1] => ([],1)
=> ? = 1 + 1
00 => 00 => [2] => ([],2)
=> ? = 1 + 1
01 => 10 => [1,1] => ([(0,1)],2)
=> ? = 2 + 1
10 => 01 => [1,1] => ([(0,1)],2)
=> ? = 1 + 1
11 => 11 => [2] => ([],2)
=> ? = 2 + 1
000 => 000 => [3] => ([],3)
=> ? = 1 + 1
001 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
010 => 100 => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
011 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
100 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
101 => 110 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
110 => 011 => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
111 => 111 => [3] => ([],3)
=> ? = 2 + 1
0000 => 0000 => [4] => ([],4)
=> ? = 1 + 1
0001 => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0010 => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0011 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0100 => 1000 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
0101 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0110 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0111 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
1000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
1001 => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
1010 => 1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
1011 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
1100 => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
1101 => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
1110 => 0111 => [1,3] => ([(2,3)],4)
=> ? = 2 + 1
1111 => 1111 => [4] => ([],4)
=> ? = 2 + 1
11000 => 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
11001 => 01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
11010 => 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
11011 => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
11100 => 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
11101 => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
11110 => 01111 => [1,4] => ([(3,4)],5)
=> ? = 2 + 1
11111 => 11111 => [5] => ([],5)
=> ? = 2 + 1
=> => [] => ?
=> ? = 1 + 1
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St001603
Mp00097: Binary words delta morphismInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001603: Integer partitions ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 50%
Values
0 => [1] => [1]
=> []
=> ? = 1 - 1
1 => [1] => [1]
=> []
=> ? = 1 - 1
00 => [2] => [2]
=> []
=> ? = 1 - 1
01 => [1,1] => [1,1]
=> [1]
=> ? = 2 - 1
10 => [1,1] => [1,1]
=> [1]
=> ? = 1 - 1
11 => [2] => [2]
=> []
=> ? = 2 - 1
000 => [3] => [3]
=> []
=> ? = 1 - 1
001 => [2,1] => [2,1]
=> [1]
=> ? = 2 - 1
010 => [1,1,1] => [1,1,1]
=> [1,1]
=> ? = 1 - 1
011 => [1,2] => [2,1]
=> [1]
=> ? = 2 - 1
100 => [1,2] => [2,1]
=> [1]
=> ? = 1 - 1
101 => [1,1,1] => [1,1,1]
=> [1,1]
=> ? = 2 - 1
110 => [2,1] => [2,1]
=> [1]
=> ? = 1 - 1
111 => [3] => [3]
=> []
=> ? = 2 - 1
0000 => [4] => [4]
=> []
=> ? = 1 - 1
0001 => [3,1] => [3,1]
=> [1]
=> ? = 2 - 1
0010 => [2,1,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
0011 => [2,2] => [2,2]
=> [2]
=> ? = 2 - 1
0100 => [1,1,2] => [2,1,1]
=> [1,1]
=> ? = 1 - 1
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
0110 => [1,2,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
0111 => [1,3] => [3,1]
=> [1]
=> ? = 2 - 1
1000 => [1,3] => [3,1]
=> [1]
=> ? = 1 - 1
1001 => [1,2,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
1011 => [1,1,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
1100 => [2,2] => [2,2]
=> [2]
=> ? = 1 - 1
1101 => [2,1,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 1
1110 => [3,1] => [3,1]
=> [1]
=> ? = 2 - 1
1111 => [4] => [4]
=> []
=> ? = 2 - 1
11000 => [2,3] => [3,2]
=> [2]
=> ? = 1 - 1
11001 => [2,2,1] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
11010 => [2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
11011 => [2,1,2] => [2,2,1]
=> [2,1]
=> 1 = 2 - 1
11100 => [3,2] => [3,2]
=> [2]
=> ? = 1 - 1
11101 => [3,1,1] => [3,1,1]
=> [1,1]
=> ? = 2 - 1
11110 => [4,1] => [4,1]
=> [1]
=> ? = 2 - 1
11111 => [5] => [5]
=> []
=> ? = 2 - 1
=> [] => ?
=> ?
=> ? = 1 - 1
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: St001604
Mp00097: Binary words delta morphismInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001604: Integer partitions ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 50%
Values
0 => [1] => [1]
=> []
=> ? = 1 - 2
1 => [1] => [1]
=> []
=> ? = 1 - 2
00 => [2] => [2]
=> []
=> ? = 1 - 2
01 => [1,1] => [1,1]
=> [1]
=> ? = 2 - 2
10 => [1,1] => [1,1]
=> [1]
=> ? = 1 - 2
11 => [2] => [2]
=> []
=> ? = 2 - 2
000 => [3] => [3]
=> []
=> ? = 1 - 2
001 => [2,1] => [2,1]
=> [1]
=> ? = 2 - 2
010 => [1,1,1] => [1,1,1]
=> [1,1]
=> ? = 1 - 2
011 => [1,2] => [2,1]
=> [1]
=> ? = 2 - 2
100 => [1,2] => [2,1]
=> [1]
=> ? = 1 - 2
101 => [1,1,1] => [1,1,1]
=> [1,1]
=> ? = 2 - 2
110 => [2,1] => [2,1]
=> [1]
=> ? = 1 - 2
111 => [3] => [3]
=> []
=> ? = 2 - 2
0000 => [4] => [4]
=> []
=> ? = 1 - 2
0001 => [3,1] => [3,1]
=> [1]
=> ? = 2 - 2
0010 => [2,1,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
0011 => [2,2] => [2,2]
=> [2]
=> ? = 2 - 2
0100 => [1,1,2] => [2,1,1]
=> [1,1]
=> ? = 1 - 2
0101 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
0110 => [1,2,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
0111 => [1,3] => [3,1]
=> [1]
=> ? = 2 - 2
1000 => [1,3] => [3,1]
=> [1]
=> ? = 1 - 2
1001 => [1,2,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
1010 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
1011 => [1,1,2] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
1100 => [2,2] => [2,2]
=> [2]
=> ? = 1 - 2
1101 => [2,1,1] => [2,1,1]
=> [1,1]
=> ? = 2 - 2
1110 => [3,1] => [3,1]
=> [1]
=> ? = 2 - 2
1111 => [4] => [4]
=> []
=> ? = 2 - 2
11000 => [2,3] => [3,2]
=> [2]
=> ? = 1 - 2
11001 => [2,2,1] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
11010 => [2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> 0 = 2 - 2
11011 => [2,1,2] => [2,2,1]
=> [2,1]
=> 0 = 2 - 2
11100 => [3,2] => [3,2]
=> [2]
=> ? = 1 - 2
11101 => [3,1,1] => [3,1,1]
=> [1,1]
=> ? = 2 - 2
11110 => [4,1] => [4,1]
=> [1]
=> ? = 2 - 2
11111 => [5] => [5]
=> []
=> ? = 2 - 2
=> [] => ?
=> ?
=> ? = 1 - 2
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: St001630
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001630: Lattices ⟶ ℤResult quality: 10% values known / values provided: 10%distinct values known / distinct values provided: 50%
Values
0 => [2] => [[2],[]]
=> ([],1)
=> ? = 1 - 1
1 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 1 - 1
00 => [3] => [[3],[]]
=> ([],1)
=> ? = 1 - 1
01 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 2 - 1
10 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 1
11 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
000 => [4] => [[4],[]]
=> ([],1)
=> ? = 1 - 1
001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
010 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? = 1 - 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? = 2 - 1
100 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? = 1 - 1
101 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 1
110 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? = 1 - 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
0000 => [5] => [[5],[]]
=> ([],1)
=> ? = 1 - 1
0001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? = 2 - 1
0010 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? = 2 - 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? = 2 - 1
0100 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? = 1 - 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? = 2 - 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? = 2 - 1
1000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? = 1 - 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? = 2 - 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? = 2 - 1
1100 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? = 1 - 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
11000 => [1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ? = 1 - 1
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ? = 2 - 1
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
11100 => [1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ? = 1 - 1
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 1
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
=> [1] => [[1],[]]
=> ([],1)
=> ? = 1 - 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001875
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001875: Lattices ⟶ ℤResult quality: 10% values known / values provided: 10%distinct values known / distinct values provided: 50%
Values
0 => [2] => [[2],[]]
=> ([],1)
=> ? = 1 + 1
1 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 1 + 1
00 => [3] => [[3],[]]
=> ([],1)
=> ? = 1 + 1
01 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 2 + 1
10 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 + 1
11 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
000 => [4] => [[4],[]]
=> ([],1)
=> ? = 1 + 1
001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 + 1
010 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? = 1 + 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? = 2 + 1
100 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? = 1 + 1
101 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? = 2 + 1
110 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? = 1 + 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
0000 => [5] => [[5],[]]
=> ([],1)
=> ? = 1 + 1
0001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? = 2 + 1
0010 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? = 2 + 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? = 2 + 1
0100 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? = 1 + 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? = 2 + 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? = 2 + 1
1000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? = 1 + 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? = 2 + 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? = 2 + 1
1100 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? = 1 + 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 + 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
11000 => [1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ? = 1 + 1
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ? = 2 + 1
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
11100 => [1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ? = 1 + 1
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 + 1
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 + 1
=> [1] => [[1],[]]
=> ([],1)
=> ? = 1 + 1
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001878
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001878: Lattices ⟶ ℤResult quality: 10% values known / values provided: 10%distinct values known / distinct values provided: 50%
Values
0 => [2] => [[2],[]]
=> ([],1)
=> ? = 1 - 1
1 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 1 - 1
00 => [3] => [[3],[]]
=> ([],1)
=> ? = 1 - 1
01 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 2 - 1
10 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 1
11 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
000 => [4] => [[4],[]]
=> ([],1)
=> ? = 1 - 1
001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
010 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? = 1 - 1
011 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? = 2 - 1
100 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? = 1 - 1
101 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 1
110 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? = 1 - 1
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
0000 => [5] => [[5],[]]
=> ([],1)
=> ? = 1 - 1
0001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? = 2 - 1
0010 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? = 2 - 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? = 2 - 1
0100 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? = 1 - 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? = 2 - 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? = 2 - 1
1000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? = 1 - 1
1001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? = 2 - 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? = 2 - 1
1100 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? = 1 - 1
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
11000 => [1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ? = 1 - 1
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ? = 2 - 1
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1 = 2 - 1
11100 => [1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ? = 1 - 1
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 1
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 1
=> [1] => [[1],[]]
=> ([],1)
=> ? = 1 - 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001876
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001876: Lattices ⟶ ℤResult quality: 10% values known / values provided: 10%distinct values known / distinct values provided: 50%
Values
0 => [2] => [[2],[]]
=> ([],1)
=> ? = 1 - 2
1 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 1 - 2
00 => [3] => [[3],[]]
=> ([],1)
=> ? = 1 - 2
01 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 2 - 2
10 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 2
11 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
000 => [4] => [[4],[]]
=> ([],1)
=> ? = 1 - 2
001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 2
010 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? = 1 - 2
011 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? = 2 - 2
100 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? = 1 - 2
101 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 2
110 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? = 1 - 2
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
0000 => [5] => [[5],[]]
=> ([],1)
=> ? = 1 - 2
0001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? = 2 - 2
0010 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? = 2 - 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? = 2 - 2
0100 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? = 1 - 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? = 2 - 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? = 2 - 2
1000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? = 1 - 2
1001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? = 2 - 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? = 2 - 2
1100 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? = 1 - 2
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
11000 => [1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ? = 1 - 2
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ? = 2 - 2
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
11100 => [1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ? = 1 - 2
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 2
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
=> [1] => [[1],[]]
=> ([],1)
=> ? = 1 - 2
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001877: Lattices ⟶ ℤResult quality: 10% values known / values provided: 10%distinct values known / distinct values provided: 50%
Values
0 => [2] => [[2],[]]
=> ([],1)
=> ? = 1 - 2
1 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 1 - 2
00 => [3] => [[3],[]]
=> ([],1)
=> ? = 1 - 2
01 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 2 - 2
10 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 2
11 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
000 => [4] => [[4],[]]
=> ([],1)
=> ? = 1 - 2
001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 2
010 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? = 1 - 2
011 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? = 2 - 2
100 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? = 1 - 2
101 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 2
110 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? = 1 - 2
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
0000 => [5] => [[5],[]]
=> ([],1)
=> ? = 1 - 2
0001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? = 2 - 2
0010 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? = 2 - 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? = 2 - 2
0100 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? = 1 - 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? = 2 - 2
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? = 2 - 2
1000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? = 1 - 2
1001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? = 2 - 2
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? = 2 - 2
1100 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? = 1 - 2
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
11000 => [1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ? = 1 - 2
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ? = 2 - 2
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 2 - 2
11100 => [1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ? = 1 - 2
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ? = 2 - 2
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ? = 2 - 2
=> [1] => [[1],[]]
=> ([],1)
=> ? = 1 - 2
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001629
Mp00097: Binary words delta morphismInteger compositions
Mp00133: Integer compositions delta morphismInteger compositions
Mp00133: Integer compositions delta morphismInteger compositions
St001629: Integer compositions ⟶ ℤResult quality: 8% values known / values provided: 8%distinct values known / distinct values provided: 50%
Values
0 => [1] => [1] => [1] => ? = 1 - 1
1 => [1] => [1] => [1] => ? = 1 - 1
00 => [2] => [1] => [1] => ? = 1 - 1
01 => [1,1] => [2] => [1] => ? = 2 - 1
10 => [1,1] => [2] => [1] => ? = 1 - 1
11 => [2] => [1] => [1] => ? = 2 - 1
000 => [3] => [1] => [1] => ? = 1 - 1
001 => [2,1] => [1,1] => [2] => ? = 2 - 1
010 => [1,1,1] => [3] => [1] => ? = 1 - 1
011 => [1,2] => [1,1] => [2] => ? = 2 - 1
100 => [1,2] => [1,1] => [2] => ? = 1 - 1
101 => [1,1,1] => [3] => [1] => ? = 2 - 1
110 => [2,1] => [1,1] => [2] => ? = 1 - 1
111 => [3] => [1] => [1] => ? = 2 - 1
0000 => [4] => [1] => [1] => ? = 1 - 1
0001 => [3,1] => [1,1] => [2] => ? = 2 - 1
0010 => [2,1,1] => [1,2] => [1,1] => ? = 2 - 1
0011 => [2,2] => [2] => [1] => ? = 2 - 1
0100 => [1,1,2] => [2,1] => [1,1] => ? = 1 - 1
0101 => [1,1,1,1] => [4] => [1] => ? = 2 - 1
0110 => [1,2,1] => [1,1,1] => [3] => 1 = 2 - 1
0111 => [1,3] => [1,1] => [2] => ? = 2 - 1
1000 => [1,3] => [1,1] => [2] => ? = 1 - 1
1001 => [1,2,1] => [1,1,1] => [3] => 1 = 2 - 1
1010 => [1,1,1,1] => [4] => [1] => ? = 2 - 1
1011 => [1,1,2] => [2,1] => [1,1] => ? = 2 - 1
1100 => [2,2] => [2] => [1] => ? = 1 - 1
1101 => [2,1,1] => [1,2] => [1,1] => ? = 2 - 1
1110 => [3,1] => [1,1] => [2] => ? = 2 - 1
1111 => [4] => [1] => [1] => ? = 2 - 1
11000 => [2,3] => [1,1] => [2] => ? = 1 - 1
11001 => [2,2,1] => [2,1] => [1,1] => ? = 2 - 1
11010 => [2,1,1,1] => [1,3] => [1,1] => ? = 2 - 1
11011 => [2,1,2] => [1,1,1] => [3] => 1 = 2 - 1
11100 => [3,2] => [1,1] => [2] => ? = 1 - 1
11101 => [3,1,1] => [1,2] => [1,1] => ? = 2 - 1
11110 => [4,1] => [1,1] => [2] => ? = 2 - 1
11111 => [5] => [1] => [1] => ? = 2 - 1
=> [] => ? => ? => ? = 1 - 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.