Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St001958
(load all 20 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => 0
[[1,2]]
 => [1,2] => 1
[[1],[2]]
 => [2,1] => 1
[[1,2,3]]
 => [1,2,3] => 1
[[1,3],[2]]
 => [2,1,3] => 2
[[1,2],[3]]
 => [3,1,2] => 2
[[1],[2],[3]]
 => [3,2,1] => 1
[[1,2,3,4]]
 => [1,2,3,4] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => 3
[[1,2,4],[3]]
 => [3,1,2,4] => 3
[[1,2,3],[4]]
 => [4,1,2,3] => 3
[[1,3],[2,4]]
 => [2,4,1,3] => 3
[[1,2],[3,4]]
 => [3,4,1,2] => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => 3
[[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => 4
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => 4
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => 4
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => 4
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => 4
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => 4
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => 4
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 4
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => 4
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => 4
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 4
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 4
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => 4
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 4
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => 4
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 4
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => 4
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => 3
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 4
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 5
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => 5
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => 5
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => 5
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => 5
Description
The degree of the polynomial interpolating the values of a permutation. Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$. This statistic records the degree of $p$.
Matching statistic: St001629
(load all 18 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[[1]]
 => [1] => [1] => ? = 0 - 4
[[1,2]]
 => [2] => [1] => ? = 1 - 4
[[1],[2]]
 => [2] => [1] => ? = 1 - 4
[[1,2,3]]
 => [3] => [1] => ? = 1 - 4
[[1,3],[2]]
 => [2,1] => [1,1] => ? = 2 - 4
[[1,2],[3]]
 => [3] => [1] => ? = 2 - 4
[[1],[2],[3]]
 => [3] => [1] => ? = 1 - 4
[[1,2,3,4]]
 => [4] => [1] => ? = 1 - 4
[[1,3,4],[2]]
 => [2,2] => [2] => ? = 3 - 4
[[1,2,4],[3]]
 => [3,1] => [1,1] => ? = 3 - 4
[[1,2,3],[4]]
 => [4] => [1] => ? = 3 - 4
[[1,3],[2,4]]
 => [2,2] => [2] => ? = 3 - 4
[[1,2],[3,4]]
 => [3,1] => [1,1] => ? = 3 - 4
[[1,4],[2],[3]]
 => [3,1] => [1,1] => ? = 3 - 4
[[1,3],[2],[4]]
 => [2,2] => [2] => ? = 3 - 4
[[1,2],[3],[4]]
 => [4] => [1] => ? = 3 - 4
[[1],[2],[3],[4]]
 => [4] => [1] => ? = 1 - 4
[[1,2,3,4,5]]
 => [5] => [1] => ? = 1 - 4
[[1,3,4,5],[2]]
 => [2,3] => [1,1] => ? = 4 - 4
[[1,2,4,5],[3]]
 => [3,2] => [1,1] => ? = 3 - 4
[[1,2,3,5],[4]]
 => [4,1] => [1,1] => ? = 4 - 4
[[1,2,3,4],[5]]
 => [5] => [1] => ? = 4 - 4
[[1,3,5],[2,4]]
 => [2,2,1] => [2,1] => 0 = 4 - 4
[[1,2,5],[3,4]]
 => [3,2] => [1,1] => ? = 4 - 4
[[1,3,4],[2,5]]
 => [2,3] => [1,1] => ? = 4 - 4
[[1,2,4],[3,5]]
 => [3,2] => [1,1] => ? = 4 - 4
[[1,2,3],[4,5]]
 => [4,1] => [1,1] => ? = 4 - 4
[[1,4,5],[2],[3]]
 => [3,2] => [1,1] => ? = 4 - 4
[[1,3,5],[2],[4]]
 => [2,2,1] => [2,1] => 0 = 4 - 4
[[1,2,5],[3],[4]]
 => [4,1] => [1,1] => ? = 4 - 4
[[1,3,4],[2],[5]]
 => [2,3] => [1,1] => ? = 4 - 4
[[1,2,4],[3],[5]]
 => [3,2] => [1,1] => ? = 4 - 4
[[1,2,3],[4],[5]]
 => [5] => [1] => ? = 4 - 4
[[1,4],[2,5],[3]]
 => [3,2] => [1,1] => ? = 4 - 4
[[1,3],[2,5],[4]]
 => [2,2,1] => [2,1] => 0 = 4 - 4
[[1,2],[3,5],[4]]
 => [4,1] => [1,1] => ? = 4 - 4
[[1,3],[2,4],[5]]
 => [2,3] => [1,1] => ? = 4 - 4
[[1,2],[3,4],[5]]
 => [3,2] => [1,1] => ? = 4 - 4
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1] => ? = 4 - 4
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1] => ? = 4 - 4
[[1,3],[2],[4],[5]]
 => [2,3] => [1,1] => ? = 3 - 4
[[1,2],[3],[4],[5]]
 => [5] => [1] => ? = 4 - 4
[[1],[2],[3],[4],[5]]
 => [5] => [1] => ? = 1 - 4
[[1,2,3,4,5,6]]
 => [6] => [1] => ? = 1 - 4
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1] => ? = 5 - 4
[[1,2,4,5,6],[3]]
 => [3,3] => [2] => ? = 5 - 4
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1] => ? = 5 - 4
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1] => ? = 5 - 4
[[1,2,3,4,5],[6]]
 => [6] => [1] => ? = 5 - 4
[[1,3,5,6],[2,4]]
 => [2,2,2] => [3] => 1 = 5 - 4
[[1,2,5,6],[3,4]]
 => [3,3] => [2] => ? = 5 - 4
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,1,1] => 1 = 5 - 4
[[1,2,4,6],[3,5]]
 => [3,2,1] => [1,1,1] => 1 = 5 - 4
[[1,2,3,6],[4,5]]
 => [4,2] => [1,1] => ? = 5 - 4
[[1,3,4,5],[2,6]]
 => [2,4] => [1,1] => ? = 5 - 4
[[1,2,4,5],[3,6]]
 => [3,3] => [2] => ? = 5 - 4
[[1,3,5,6],[2],[4]]
 => [2,2,2] => [3] => 1 = 5 - 4
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,1,1] => 1 = 5 - 4
[[1,2,4,6],[3],[5]]
 => [3,2,1] => [1,1,1] => 1 = 5 - 4
[[1,3,5],[2,4,6]]
 => [2,2,2] => [3] => 1 = 5 - 4
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,1,1] => 1 = 5 - 4
[[1,2,4],[3,5,6]]
 => [3,2,1] => [1,1,1] => 1 = 5 - 4
[[1,4,6],[2,5],[3]]
 => [3,2,1] => [1,1,1] => 1 = 5 - 4
[[1,3,6],[2,5],[4]]
 => [2,2,2] => [3] => 1 = 5 - 4
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,1,1] => 1 = 5 - 4
[[1,2,6],[3,4],[5]]
 => [3,2,1] => [1,1,1] => 1 = 5 - 4
[[1,3,5],[2,6],[4]]
 => [2,2,2] => [3] => 1 = 5 - 4
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,1,1] => 1 = 5 - 4
[[1,2,4],[3,6],[5]]
 => [3,2,1] => [1,1,1] => 1 = 5 - 4
[[1,3,5],[2,4],[6]]
 => [2,2,2] => [3] => 1 = 5 - 4
[[1,4,6],[2],[3],[5]]
 => [3,2,1] => [1,1,1] => 1 = 5 - 4
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,1,1] => 1 = 5 - 4
[[1,3,5],[2],[4],[6]]
 => [2,2,2] => [3] => 1 = 5 - 4
[[1,3],[2,5],[4,6]]
 => [2,2,2] => [3] => 1 = 5 - 4
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,1,1] => 1 = 5 - 4
[[1,2],[3,4],[5,6]]
 => [3,2,1] => [1,1,1] => 1 = 5 - 4
[[1,4],[2,6],[3],[5]]
 => [3,2,1] => [1,1,1] => 1 = 5 - 4
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,1,1] => 1 = 5 - 4
[[1,3],[2,5],[4],[6]]
 => [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: St001633
(load all 3 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001633: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => => ?
 => ? = 0 - 1
[[1,2]]
 => 0 => ([(0,1)],2)
 => 0 = 1 - 1
[[1],[2]]
 => 1 => ([(0,1)],2)
 => 0 = 1 - 1
[[1,2,3]]
 => 00 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[1,3],[2]]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[1,2],[3]]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[1],[2],[3]]
 => 11 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[1,2,3,4]]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[1,3,4],[2]]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1,2,4],[3]]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,2,3],[4]]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1,3],[2,4]]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,2],[3,4]]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,4],[2],[3]]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1,3],[2],[4]]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,2],[3],[4]]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[1,2,3,4,5]]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1,2,4,5],[3]]
 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 3 - 1
[[1,2,3,5],[4]]
 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,2,3,4],[5]]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1,3,5],[2,4]]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,5],[3,4]]
 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,3,4],[2,5]]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,2,4],[3,5]]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,3],[4,5]]
 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,4,5],[2],[3]]
 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[[1,3,5],[2],[4]]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,5],[3],[4]]
 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,3,4],[2],[5]]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,2,4],[3],[5]]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,3],[4],[5]]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[[1,4],[2,5],[3]]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,3],[2,5],[4]]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2],[3,5],[4]]
 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,3],[2,4],[5]]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,2],[3,4],[5]]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,5],[2],[3],[4]]
 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1,4],[2],[3],[5]]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,3],[2],[4],[5]]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 3 - 1
[[1,2],[3],[4],[5]]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1],[2],[3],[4],[5]]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[1,2,3,4,5,6]]
 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 - 1
[[1,2,4,5,6],[3]]
 => 01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,2,3,5,6],[4]]
 => 00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,4,6],[5]]
 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,2,3,4,5],[6]]
 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 - 1
[[1,3,5,6],[2,4]]
 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,5,6],[3,4]]
 => 01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,3,4,6],[2,5]]
 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,6],[3,5]]
 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 - 1
[[1,2,3,6],[4,5]]
 => 00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5 - 1
[[1,3,4,5],[2,6]]
 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,5],[3,6]]
 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,5],[4,6]]
 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,4],[5,6]]
 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,4,5,6],[2],[3]]
 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,3,5,6],[2],[4]]
 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,5,6],[3],[4]]
 => 01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 - 1
[[1,3,4,6],[2],[5]]
 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,6],[3],[5]]
 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 - 1
[[1,2,3,6],[4],[5]]
 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 - 1
[[1,3,4,5],[2],[6]]
 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,5],[3],[6]]
 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,5],[4],[6]]
 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,4],[5],[6]]
 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,3,5],[2,4,6]]
 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 - 1
[[1],[2],[3],[4],[5],[6]]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[1,2,3,4,5,6,7]]
 => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0 = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000848
(load all 4 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000848: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => => ?
 => ? = 0 - 1
[[1,2]]
 => 0 => ([(0,1)],2)
 => 0 = 1 - 1
[[1],[2]]
 => 1 => ([(0,1)],2)
 => 0 = 1 - 1
[[1,2,3]]
 => 00 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[1,3],[2]]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[1,2],[3]]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[1],[2],[3]]
 => 11 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[1,2,3,4]]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[1,3,4],[2]]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1,2,4],[3]]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,2,3],[4]]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1,3],[2,4]]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,2],[3,4]]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,4],[2],[3]]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1,3],[2],[4]]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,2],[3],[4]]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[1,2,3,4,5]]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1,2,4,5],[3]]
 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 3 - 1
[[1,2,3,5],[4]]
 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,2,3,4],[5]]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1,3,5],[2,4]]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,5],[3,4]]
 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,3,4],[2,5]]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,2,4],[3,5]]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,3],[4,5]]
 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,4,5],[2],[3]]
 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[[1,3,5],[2],[4]]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,5],[3],[4]]
 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,3,4],[2],[5]]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,2,4],[3],[5]]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,3],[4],[5]]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[[1,4],[2,5],[3]]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,3],[2,5],[4]]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2],[3,5],[4]]
 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,3],[2,4],[5]]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,2],[3,4],[5]]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,5],[2],[3],[4]]
 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1,4],[2],[3],[5]]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,3],[2],[4],[5]]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 3 - 1
[[1,2],[3],[4],[5]]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1],[2],[3],[4],[5]]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[1,2,3,4,5,6]]
 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 - 1
[[1,2,4,5,6],[3]]
 => 01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,2,3,5,6],[4]]
 => 00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,4,6],[5]]
 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,2,3,4,5],[6]]
 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 - 1
[[1,3,5,6],[2,4]]
 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,5,6],[3,4]]
 => 01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,3,4,6],[2,5]]
 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,6],[3,5]]
 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 - 1
[[1,2,3,6],[4,5]]
 => 00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5 - 1
[[1,3,4,5],[2,6]]
 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,5],[3,6]]
 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,5],[4,6]]
 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,4],[5,6]]
 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,4,5,6],[2],[3]]
 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,3,5,6],[2],[4]]
 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,5,6],[3],[4]]
 => 01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 - 1
[[1,3,4,6],[2],[5]]
 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,6],[3],[5]]
 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 - 1
[[1,2,3,6],[4],[5]]
 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 - 1
[[1,3,4,5],[2],[6]]
 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,5],[3],[6]]
 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,5],[4],[6]]
 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,4],[5],[6]]
 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,3,5],[2,4,6]]
 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 - 1
[[1],[2],[3],[4],[5],[6]]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
Description
The balance constant multiplied with the number of linear extensions of a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion $P(x,y)$ of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. The balance constant of a poset is $\max\min(P(x,y), P(y,x)).$ Kislitsyn [1] conjectured that every poset which is not a chain is $1/3$-balanced. Brightwell, Felsner and Trotter [2] show that it is at least $(1-\sqrt 5)/10$-balanced. Olson and Sagan [3] exhibit various posets that are $1/2$-balanced.
Matching statistic: St000849
(load all 4 compositions to match this statistic)
Mapping
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000849: Posets ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[[1]]
 => => ?
 => ? = 0 - 1
[[1,2]]
 => 0 => ([(0,1)],2)
 => 0 = 1 - 1
[[1],[2]]
 => 1 => ([(0,1)],2)
 => 0 = 1 - 1
[[1,2,3]]
 => 00 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[1,3],[2]]
 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[1,2],[3]]
 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[1],[2],[3]]
 => 11 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[[1,2,3,4]]
 => 000 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[1,3,4],[2]]
 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1,2,4],[3]]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,2,3],[4]]
 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1,3],[2,4]]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,2],[3,4]]
 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,4],[2],[3]]
 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1,3],[2],[4]]
 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[[1,2],[3],[4]]
 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => 111 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[[1,2,3,4,5]]
 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1,2,4,5],[3]]
 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 3 - 1
[[1,2,3,5],[4]]
 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,2,3,4],[5]]
 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1,3,5],[2,4]]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,5],[3,4]]
 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,3,4],[2,5]]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,2,4],[3,5]]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,3],[4,5]]
 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,4,5],[2],[3]]
 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[[1,3,5],[2],[4]]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,5],[3],[4]]
 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,3,4],[2],[5]]
 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,2,4],[3],[5]]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2,3],[4],[5]]
 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[[1,4],[2,5],[3]]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,3],[2,5],[4]]
 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,2],[3,5],[4]]
 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 4 - 1
[[1,3],[2,4],[5]]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,2],[3,4],[5]]
 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ? = 4 - 1
[[1,5],[2],[3],[4]]
 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1,4],[2],[3],[5]]
 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[[1,3],[2],[4],[5]]
 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 3 - 1
[[1,2],[3],[4],[5]]
 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[[1],[2],[3],[4],[5]]
 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[[1,2,3,4,5,6]]
 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 - 1
[[1,2,4,5,6],[3]]
 => 01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,2,3,5,6],[4]]
 => 00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,4,6],[5]]
 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,2,3,4,5],[6]]
 => 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 - 1
[[1,3,5,6],[2,4]]
 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,5,6],[3,4]]
 => 01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,3,4,6],[2,5]]
 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,6],[3,5]]
 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 - 1
[[1,2,3,6],[4,5]]
 => 00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
 => ? = 5 - 1
[[1,3,4,5],[2,6]]
 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,5],[3,6]]
 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,5],[4,6]]
 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,4],[5,6]]
 => 00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,4,5,6],[2],[3]]
 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,3,5,6],[2],[4]]
 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,5,6],[3],[4]]
 => 01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 - 1
[[1,3,4,6],[2],[5]]
 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,6],[3],[5]]
 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 - 1
[[1,2,3,6],[4],[5]]
 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 5 - 1
[[1,3,4,5],[2],[6]]
 => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,4,5],[3],[6]]
 => 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,5],[4],[6]]
 => 00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
 => ? = 5 - 1
[[1,2,3,4],[5],[6]]
 => 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[[1,3,5],[2,4,6]]
 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
 => ? = 5 - 1
[[1],[2],[3],[4],[5],[6]]
 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
Description
The number of 1/3-balanced pairs in a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains. Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a $1/3$-balanced pair.
Matching statistic: St001060
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 33%
Values
[[1]]
 => [1] => [1] => ([],1)
 => ? = 0 - 2
[[1,2]]
 => [2] => [1] => ([],1)
 => ? = 1 - 2
[[1],[2]]
 => [2] => [1] => ([],1)
 => ? = 1 - 2
[[1,2,3]]
 => [3] => [1] => ([],1)
 => ? = 1 - 2
[[1,3],[2]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 2
[[1,2],[3]]
 => [3] => [1] => ([],1)
 => ? = 2 - 2
[[1],[2],[3]]
 => [3] => [1] => ([],1)
 => ? = 1 - 2
[[1,2,3,4]]
 => [4] => [1] => ([],1)
 => ? = 1 - 2
[[1,3,4],[2]]
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
[[1,2,4],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,2,3],[4]]
 => [4] => [1] => ([],1)
 => ? = 3 - 2
[[1,3],[2,4]]
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
[[1,2],[3,4]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,4],[2],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,3],[2],[4]]
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
[[1,2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? = 3 - 2
[[1],[2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? = 1 - 2
[[1,2,3,4,5]]
 => [5] => [1] => ([],1)
 => ? = 1 - 2
[[1,3,4,5],[2]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,4,5],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,2,3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,3,4],[5]]
 => [5] => [1] => ([],1)
 => ? = 4 - 2
[[1,3,5],[2,4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
[[1,2,5],[3,4]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3,4],[2,5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,4],[3,5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,3],[4,5]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,4,5],[2],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
[[1,2,5],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3,4],[2],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,4],[3],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? = 4 - 2
[[1,4],[2,5],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
[[1,2],[3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3],[2,4],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2],[3,4],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3],[2],[4],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? = 4 - 2
[[1],[2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? = 1 - 2
[[1,2,3,4,5,6]]
 => [6] => [1] => ([],1)
 => ? = 1 - 2
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 5 - 2
[[1,2,4,5,6],[3]]
 => [3,3] => [2] => ([],2)
 => ? = 5 - 2
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1] => ([(0,1)],2)
 => ? = 5 - 2
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? = 5 - 2
[[1,2,3,4,5],[6]]
 => [6] => [1] => ([],1)
 => ? = 5 - 2
[[1,3,5,6],[2,4]]
 => [2,2,2] => [3] => ([],3)
 => ? = 5 - 2
[[1,2,5,6],[3,4]]
 => [3,3] => [2] => ([],2)
 => ? = 5 - 2
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,4,6],[3,5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,3,6],[4,5]]
 => [4,2] => [1,1] => ([(0,1)],2)
 => ? = 5 - 2
[[1,3,4,5],[2,6]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 5 - 2
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,4,6],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,4],[3,5,6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,4,6],[2,5],[3]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,6],[3,4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,4],[3,6],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,4,6],[2],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2],[3,4],[5,6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,4],[2,6],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
Description
The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [1] => [1,0]
 => 0
[[1,2]]
 => [2] => [1,1] => [1,0,1,0]
 => 1
[[1],[2]]
 => [2] => [1,1] => [1,0,1,0]
 => 1
[[1,2,3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => ? = 1
[[1,3],[2]]
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 2
[[1,2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => ? = 2
[[1],[2],[3]]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => ? = 1
[[1,2,3,4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => ? = 1
[[1,3,4],[2]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[[1,2,4],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => ? = 3
[[1,2,3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => ? = 3
[[1,3],[2,4]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[[1,2],[3,4]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => ? = 3
[[1,4],[2],[3]]
 => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => ? = 3
[[1,3],[2],[4]]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[[1,2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => ? = 3
[[1],[2],[3],[4]]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => ? = 1
[[1,2,3,4,5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[[1,3,4,5],[2]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 4
[[1,2,4,5],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[[1,2,3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[[1,2,3,4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[[1,3,5],[2,4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[1,2,5],[3,4]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[1,3,4],[2,5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 4
[[1,2,4],[3,5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3],[4,5]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[[1,4,5],[2],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[1,3,5],[2],[4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[1,2,5],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[[1,3,4],[2],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 4
[[1,2,4],[3],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[[1,4],[2,5],[3]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[1,3],[2,5],[4]]
 => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4
[[1,2],[3,5],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[[1,3],[2,4],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 4
[[1,2],[3,4],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[[1,3],[2],[4],[5]]
 => [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 3
[[1,2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[[1],[2],[3],[4],[5]]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[[1,2,3,4,5,6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[[1,3,4,5,6],[2]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5
[[1,2,4,5,6],[3]]
 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 5
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[[1,2,3,4,5],[6]]
 => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5
[[1,3,5,6],[2,4]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[[1,2,5,6],[3,4]]
 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 5
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 5
[[1,2,4,6],[3,5]]
 => [3,2,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 5
[[1,2,3,6],[4,5]]
 => [4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5
[[1,3,4,5],[2,6]]
 => [2,4] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 5
[[1,2,4,5],[3,6]]
 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 5
[[1,2,3,5],[4,6]]
 => [4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5
[[1,2,3,4],[5,6]]
 => [5,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 5
[[1,4,5,6],[2],[3]]
 => [3,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 5
[[1,3,5,6],[2],[4]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[[1,2,5,6],[3],[4]]
 => [4,2] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 5
[[1,3,5],[2,4,6]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[[1,3,6],[2,5],[4]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[[1,3,5],[2,6],[4]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[[1,3,5],[2,4],[6]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[[1,3,5],[2],[4],[6]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[[1,3],[2,5],[4,6]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[[1,3],[2,5],[4],[6]]
 => [2,2,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 17%
Values
[[1]]
 => [1] => [1] => ([],1)
 => ? = 0 - 2
[[1,2]]
 => [2] => [1] => ([],1)
 => ? = 1 - 2
[[1],[2]]
 => [2] => [1] => ([],1)
 => ? = 1 - 2
[[1,2,3]]
 => [3] => [1] => ([],1)
 => ? = 1 - 2
[[1,3],[2]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => ? = 2 - 2
[[1,2],[3]]
 => [3] => [1] => ([],1)
 => ? = 2 - 2
[[1],[2],[3]]
 => [3] => [1] => ([],1)
 => ? = 1 - 2
[[1,2,3,4]]
 => [4] => [1] => ([],1)
 => ? = 1 - 2
[[1,3,4],[2]]
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
[[1,2,4],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,2,3],[4]]
 => [4] => [1] => ([],1)
 => ? = 3 - 2
[[1,3],[2,4]]
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
[[1,2],[3,4]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,4],[2],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,3],[2],[4]]
 => [2,2] => [2] => ([],2)
 => ? = 3 - 2
[[1,2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? = 3 - 2
[[1],[2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? = 1 - 2
[[1,2,3,4,5]]
 => [5] => [1] => ([],1)
 => ? = 1 - 2
[[1,3,4,5],[2]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,4,5],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,2,3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,3,4],[5]]
 => [5] => [1] => ([],1)
 => ? = 4 - 2
[[1,3,5],[2,4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 2
[[1,2,5],[3,4]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3,4],[2,5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,4],[3,5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,3],[4,5]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,4,5],[2],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3,5],[2],[4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 2
[[1,2,5],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3,4],[2],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,4],[3],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2,3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? = 4 - 2
[[1,4],[2,5],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3],[2,5],[4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 4 - 2
[[1,2],[3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3],[2,4],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,2],[3,4],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? = 4 - 2
[[1,3],[2],[4],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? = 3 - 2
[[1,2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? = 4 - 2
[[1],[2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? = 1 - 2
[[1,2,3,4,5,6]]
 => [6] => [1] => ([],1)
 => ? = 1 - 2
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? = 5 - 2
[[1,2,4,5,6],[3]]
 => [3,3] => [2] => ([],2)
 => ? = 5 - 2
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1] => ([(0,1)],2)
 => ? = 5 - 2
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? = 5 - 2
[[1,2,3,4,5],[6]]
 => [6] => [1] => ([],1)
 => ? = 5 - 2
[[1,3,5,6],[2,4]]
 => [2,2,2] => [3] => ([],3)
 => ? = 5 - 2
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,4,6],[3,5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,4,6],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,4],[3,5,6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,4,6],[2,5],[3]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,6],[3,4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2,4],[3,6],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,4,6],[2],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,2],[3,4],[5,6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,4],[2,6],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 5 - 2
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 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.