Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
(load all 2 compositions to match this statistic)
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => [1] => 1 => 1
{{1,2}}
 => {{1,2}}
 => [2] => 10 => 1
{{1},{2}}
 => {{1},{2}}
 => [1,1] => 11 => 2
{{1,2,3}}
 => {{1,2,3}}
 => [3] => 100 => 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => 101 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => 110 => 2
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => 101 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => 111 => 3
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => 1000 => 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => 1001 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => 1010 => 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => 1001 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => 1011 => 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => 1010 => 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => 1100 => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => 1101 => 2
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => 1010 => 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => 1001 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => 1011 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => 1110 => 3
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => 1101 => 2
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => 1011 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => 1111 => 4
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => 10000 => 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => 10001 => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => 10010 => 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => 10001 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => 10011 => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => 10010 => 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => 10100 => 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => 10101 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => 10010 => 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => 10001 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => 10011 => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => 10110 => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => 10101 => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => 10011 => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => 10111 => 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => 10010 => 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => 10100 => 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => 10101 => 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => 11000 => 2
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => 10100 => 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => 11001 => 2
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => 11001 => 2
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => 11010 => 2
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => 10101 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => 11011 => 2
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => 10010 => 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => 10100 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000363
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => [1] => ([],1)
 => 1
{{1,2}}
 => {{1,2}}
 => [2] => ([],2)
 => 1
{{1},{2}}
 => {{1},{2}}
 => [1,1] => ([(0,1)],2)
 => 2
{{1,2,3}}
 => {{1,2,3}}
 => [3] => ([],3)
 => 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => ([(0,2),(1,2)],3)
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => ([(1,2)],3)
 => 2
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => ([(0,2),(1,2)],3)
 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => ([],4)
 => 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => ([(2,3)],4)
 => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => ([(1,3),(2,3)],4)
 => 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => ([],5)
 => 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => ([(2,4),(3,4)],5)
 => 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => ([(2,4),(3,4)],5)
 => 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => ([(3,4)],5)
 => 2
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => ([(2,4),(3,4)],5)
 => 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => ([(2,4),(3,4)],5)
 => 1
Description
The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of $G$. This is also the number of independent sets of maximal cardinality of $G$.
Matching statistic: St000382
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => [1] => [1] => 1
{{1,2}}
 => {{1,2}}
 => [2] => [1,1] => 1
{{1},{2}}
 => {{1},{2}}
 => [1,1] => [2] => 2
{{1,2,3}}
 => {{1,2,3}}
 => [3] => [1,1,1] => 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [1,2] => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [2,1] => 2
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [1,2] => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => [3] => 3
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1] => 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [1,1,2] => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => [1,2,1] => 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => [1,1,2] => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [1,3] => 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => [1,2,1] => 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => [2,1,1] => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [2,2] => 2
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => [1,2,1] => 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [1,1,2] => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [1,3] => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [3,1] => 3
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [2,2] => 2
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [1,3] => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => 4
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [1,1,1,2] => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => [1,1,1,2] => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => [1,1,1,2] => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,3] => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [1,2,2] => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,3] => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [1,2,2] => 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => [2,1,1,1] => 2
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => [1,2,1,1] => 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [2,1,2] => 2
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => [2,1,2] => 2
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [2,2,1] => 2
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => [1,2,2] => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [2,3] => 2
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => [1] => [1] => 1
{{1,2}}
 => {{1,2}}
 => [2] => [1,1] => 1
{{1},{2}}
 => {{1},{2}}
 => [1,1] => [2] => 2
{{1,2,3}}
 => {{1,2,3}}
 => [3] => [1,1,1] => 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1] => [2,1] => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,2] => [1,2] => 2
{{1},{2,3}}
 => {{1,3},{2}}
 => [2,1] => [2,1] => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => [3] => 3
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => [1,1,1,1] => 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1] => [2,1,1] => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2] => [1,2,1] => 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1] => [2,1,1] => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1] => [3,1] => 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2] => [1,2,1] => 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3] => [1,1,2] => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,2,1] => [2,2] => 2
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2] => [1,2,1] => 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1] => [2,1,1] => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1] => [3,1] => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,1,2] => [1,3] => 3
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,2,1] => [2,2] => 2
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1] => [3,1] => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => 4
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [3,2] => [1,2,1,1] => 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [4,1] => [2,1,1,1] => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [3,2] => [1,2,1,1] => 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => [1,1,2,1] => 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,2,1] => [2,2,1] => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [3,2] => [1,2,1,1] => 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1] => [2,1,1,1] => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1] => [3,1,1] => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1] => [2,2,1] => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => [3,1,1] => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1] => [4,1] => 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2] => [1,2,1,1] => 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => [1,1,2,1] => 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1] => [2,2,1] => 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,4] => [1,1,1,2] => 2
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [2,3] => [1,1,2,1] => 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,1] => [2,1,2] => 2
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => [2,1,2] => 2
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => [1,2,2] => 2
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1] => [2,2,1] => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,2,1,1] => [3,2] => 2
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,2] => [1,2,1,1] => 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => [1,1,2,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000729
(load all 11 compositions to match this statistic)
Mapping
St000729: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => ? = 1
{{1,2}}
 => 1
{{1},{2}}
 => 2
{{1,2,3}}
 => 1
{{1,2},{3}}
 => 1
{{1,3},{2}}
 => 2
{{1},{2,3}}
 => 1
{{1},{2},{3}}
 => 3
{{1,2,3,4}}
 => 1
{{1,2,3},{4}}
 => 1
{{1,2,4},{3}}
 => 1
{{1,2},{3,4}}
 => 1
{{1,2},{3},{4}}
 => 1
{{1,3,4},{2}}
 => 1
{{1,3},{2,4}}
 => 2
{{1,3},{2},{4}}
 => 2
{{1,4},{2,3}}
 => 1
{{1},{2,3,4}}
 => 1
{{1},{2,3},{4}}
 => 1
{{1,4},{2},{3}}
 => 3
{{1},{2,4},{3}}
 => 2
{{1},{2},{3,4}}
 => 1
{{1},{2},{3},{4}}
 => 4
{{1,2,3,4,5}}
 => 1
{{1,2,3,4},{5}}
 => 1
{{1,2,3,5},{4}}
 => 1
{{1,2,3},{4,5}}
 => 1
{{1,2,3},{4},{5}}
 => 1
{{1,2,4,5},{3}}
 => 1
{{1,2,4},{3,5}}
 => 1
{{1,2,4},{3},{5}}
 => 1
{{1,2,5},{3,4}}
 => 1
{{1,2},{3,4,5}}
 => 1
{{1,2},{3,4},{5}}
 => 1
{{1,2,5},{3},{4}}
 => 1
{{1,2},{3,5},{4}}
 => 1
{{1,2},{3},{4,5}}
 => 1
{{1,2},{3},{4},{5}}
 => 1
{{1,3,4,5},{2}}
 => 1
{{1,3,4},{2,5}}
 => 1
{{1,3,4},{2},{5}}
 => 1
{{1,3,5},{2,4}}
 => 2
{{1,3},{2,4,5}}
 => 1
{{1,3},{2,4},{5}}
 => 2
{{1,3,5},{2},{4}}
 => 2
{{1,3},{2,5},{4}}
 => 2
{{1,3},{2},{4,5}}
 => 1
{{1,3},{2},{4},{5}}
 => 2
{{1,4,5},{2,3}}
 => 1
{{1,4},{2,3,5}}
 => 1
{{1,4},{2,3},{5}}
 => 1
Description
The minimal arc length of a set partition. The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the minimal arc length is the size of the ground set (as the minimum of the empty set in the universe of arcs of length less than the size of the ground set).
Matching statistic: St000326
Mapping
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => [1] => => ? = 1
{{1,2}}
 => {{1,2}}
 => [2,1] => 1 => 1
{{1},{2}}
 => {{1},{2}}
 => [1,2] => 0 => 2
{{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => 10 => 1
{{1,2},{3}}
 => {{1,2},{3}}
 => [2,1,3] => 10 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,3,2] => 01 => 2
{{1},{2,3}}
 => {{1,3},{2}}
 => [3,2,1] => 11 => 1
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => 00 => 3
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => 100 => 1
{{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => 100 => 1
{{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => 101 => 1
{{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 101 => 1
{{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => 100 => 1
{{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 111 => 1
{{1,3},{2,4}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => 010 => 2
{{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => 010 => 2
{{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => 100 => 1
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 110 => 1
{{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => 110 => 1
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => 001 => 3
{{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => 011 => 2
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 110 => 1
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 000 => 4
{{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 1000 => 1
{{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => [2,3,4,1,5] => 1000 => 1
{{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => [2,3,1,5,4] => 1001 => 1
{{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => [2,3,5,4,1] => 1001 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => [2,3,1,4,5] => 1000 => 1
{{1,2,4,5},{3}}
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 1011 => 1
{{1,2,4},{3,5}}
 => {{1,2},{3,4,5}}
 => [2,1,4,5,3] => 1010 => 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => [2,1,4,3,5] => 1010 => 1
{{1,2,5},{3,4}}
 => {{1,2,4},{3,5}}
 => [2,4,5,1,3] => 1000 => 1
{{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [2,4,3,5,1] => 1010 => 1
{{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [2,4,3,1,5] => 1010 => 1
{{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => [2,1,3,5,4] => 1001 => 1
{{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,1,5,4,3] => 1011 => 1
{{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [2,5,3,4,1] => 1010 => 1
{{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => 1000 => 1
{{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [4,3,2,5,1] => 1110 => 1
{{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [4,3,5,1,2] => 1010 => 1
{{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [4,3,2,1,5] => 1110 => 1
{{1,3,5},{2,4}}
 => {{1},{2,3,4,5}}
 => [1,3,4,5,2] => 0100 => 2
{{1,3},{2,4,5}}
 => {{1,5},{2,3,4}}
 => [5,3,4,2,1] => 1110 => 1
{{1,3},{2,4},{5}}
 => {{1},{2,3,4},{5}}
 => [1,3,4,2,5] => 0100 => 2
{{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [1,3,5,4,2] => 0101 => 2
{{1,3},{2,5},{4}}
 => {{1},{2,3},{4,5}}
 => [1,3,2,5,4] => 0101 => 2
{{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [5,3,2,4,1] => 1110 => 1
{{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => 0100 => 2
{{1,4,5},{2,3}}
 => {{1,3,5},{2,4}}
 => [3,4,5,2,1] => 1100 => 1
{{1,4},{2,3,5}}
 => {{1,3},{2,4,5}}
 => [3,4,1,5,2] => 1100 => 1
{{1,4},{2,3},{5}}
 => {{1,3},{2,4},{5}}
 => [3,4,1,2,5] => 1000 => 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.