Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001092
Mapping
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => {{1}}
 => {{1}}
 => [1]
 => 0
[1,0,1,0]
 => {{1},{2}}
 => {{1},{2}}
 => [1,1]
 => 0
[1,1,0,0]
 => {{1,2}}
 => {{1,2}}
 => [2]
 => 1
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1]
 => 0
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => {{1,3},{2}}
 => [2,1]
 => 1
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => {{1,2},{3}}
 => [2,1]
 => 1
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => {{1},{2,3}}
 => [2,1]
 => 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => {{1,2,3}}
 => [3]
 => 0
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => [2,1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => [2,1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => [2,1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => [3,1]
 => 0
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => [3,1]
 => 0
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => [2,1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [2,1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => {{1,4},{2,3}}
 => [2,2]
 => 1
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => [3,1]
 => 0
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => [2,2]
 => 1
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => [2,2]
 => 1
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => [4]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => {{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => {{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => {{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => [3,1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => {{1,3,5},{2},{4}}
 => [3,1,1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => {{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => {{1,5},{2,4},{3}}
 => [2,2,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => {{1,3,4},{2},{5}}
 => [3,1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => {{1,4},{2,5},{3}}
 => [2,2,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => {{1,3},{2,5},{4}}
 => [2,2,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => {{1,3,4,5},{2}}
 => [4,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => {{1,2,5},{3},{4}}
 => [3,1,1]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => {{1,2,4},{3},{5}}
 => [3,1,1]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => {{1,2},{3,5},{4}}
 => [2,2,1]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => {{1,2,4,5},{3}}
 => [4,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => {{1,5},{2,3},{4}}
 => [2,2,1]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => {{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => {{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => {{1,5},{2},{3,4}}
 => [2,2,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => {{1,4},{2,3},{5}}
 => [2,2,1]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => {{1,4},{2},{3,5}}
 => [2,2,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => {{1},{2,3,5},{4}}
 => [3,1,1]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => {{1,4,5},{2,3}}
 => [3,2]
 => 1
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St001491
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1] => [1] => => ? = 0
[1,0,1,0]
 => [2,1] => [1,2] => 0 => ? = 0
[1,1,0,0]
 => [1,2] => [1,2] => 0 => ? = 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => 00 => ? = 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => 00 => ? = 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => 00 => ? = 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => 01 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 00 => ? = 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => 000 => ? = 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => 000 => ? = 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => 000 => ? = 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => 001 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 000 => ? = 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => 000 => ? = 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => 000 => ? = 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,4,2,3] => 010 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,4,2,3] => 010 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => 000 => ? = 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,2,4,3] => 001 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => 001 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 000 => ? = 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 0000 => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => 0000 => ? = 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => 0000 => ? = 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => 0000 => ? = 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => 0000 => ? = 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => 0000 => ? = 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => 0000 => ? = 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => 0001 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 0000 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => 0000 => ? = 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => 0000 => ? = 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => 0000 => ? = 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => 0000 => ? = 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,5,2,4,3] => 0101 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,5,2,3,4] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,3,4,5] => 0000 => ? = 0
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,2,3,4,5] => 0000 => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,2,4,3,5] => 0010 => 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [1,2,5,3,4] => 0010 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,2,5,3,4] => 0010 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [1,3,4,2,5] => 0010 => 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [1,3,5,2,4] => 0010 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [1,4,5,2,3] => 0010 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [1,4,5,2,3] => 0010 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,4,5] => 0000 => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [1,2,3,5,4] => 0001 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [1,2,4,5,3] => 0001 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [1,3,4,5,2] => 0001 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 00000 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [1,2,3,4,5,6] => 00000 => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [1,2,3,4,5,6] => 00000 => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [1,2,3,4,6,5] => 00001 => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [1,2,3,4,5,6] => 00000 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 00000 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,2,3,4,5,6] => 00000 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => [1,2,3,5,4,6] => 00010 => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,6,2,1] => [1,2,3,6,4,5] => 00010 => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,2,1] => [1,2,3,6,4,5] => 00010 => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [1,2,3,4,5,6] => 00000 => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [1,2,3,4,6,5] => 00001 => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [4,3,5,6,2,1] => [1,2,3,5,6,4] => 00001 => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,2,3,4,5,6] => 00000 => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,3,1] => [1,2,3,4,5,6] => 00000 => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [1,2,3,4,5,6] => 00000 => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,3,1] => [1,2,3,4,5,6] => 00000 => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,3,1] => [1,2,3,4,6,5] => 00001 => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,3,1] => [1,2,3,4,5,6] => 00000 => ? = 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001624
Mapping
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1,0]
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,1,0,0]
 => [1,1,0,0]
 => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 0 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
 => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => ?
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
 => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
 => ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => ?
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,5),(0,6),(1,11),(2,8),(3,7),(4,10),(5,9),(6,1),(6,9),(8,7),(9,4),(9,11),(10,3),(10,8),(11,2),(11,10)],12)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,5),(0,6),(1,10),(3,7),(4,8),(5,9),(6,1),(6,9),(7,8),(8,2),(9,3),(9,10),(10,4),(10,7)],11)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,4),(0,5),(0,6),(1,8),(1,9),(2,11),(2,15),(3,10),(3,14),(4,7),(4,13),(5,7),(5,12),(6,3),(6,12),(6,13),(7,18),(8,17),(9,17),(10,19),(11,16),(12,10),(12,18),(13,2),(13,14),(13,18),(14,11),(14,19),(15,8),(15,16),(16,17),(18,1),(18,15),(18,19),(19,9),(19,16)],20)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
 => ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
 => ? = 1 + 1
Description
The breadth of a lattice. The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.