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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000039
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000039: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,3,2] => 0
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => 0
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => 0
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,3,4,2] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,4,1,2] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => [3,1,2,4] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => [3,2,4,1] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,4,2,1] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,1,3,2] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,4,2,3] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,3,1,2] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [2,4,1,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,1,3] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,3,2,1] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,2,3,4] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,4,3,1] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,2,1,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [3,1,4,2] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,3,4,5,2] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,3,5,2,4] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,4,5,3,2] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,4,2,5,3] => 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,2,3,4,5] => 0
Description
The number of crossings of a permutation.
A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$.
Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross.
Matching statistic: St001862
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00309: Permutations —inverse toric promotion⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => [3,1,2] => 0
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => [2,3,1] => 1
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,3,4,2] => [1,3,4,2] => 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,4,1,2] => [3,4,1,2] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => [3,1,2,4] => [3,1,2,4] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => [3,2,4,1] => [3,2,4,1] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,4,2,1] => [3,4,2,1] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [4,1,3,2] => [4,1,3,2] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,4,2,3] => [1,4,2,3] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,3,1,2] => [4,3,1,2] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [2,4,1,3] => [2,4,1,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,1,3] => [4,2,1,3] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,3,2,1] => [4,3,2,1] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,4,3,1] => [2,4,3,1] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,2,1,4] => [3,2,1,4] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [3,1,4,2] => [3,1,4,2] => 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,3,4,5,2] => [1,3,4,5,2] => 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,3,5,2,4] => [1,3,5,2,4] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,4,5,3,2] => [1,4,5,3,2] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,4,2,5,3] => [1,4,2,5,3] => 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
* $i < j \leq \pi(i) < \pi(j)$, or
* $-i < j \leq -\pi(i) < \pi(j)$, or
* $i > j > \pi(i) > \pi(j)$.
Matching statistic: St001857
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 67%
Values
{{1}}
=> {{1}}
=> [1] => [1] => 0
{{1,2}}
=> {{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
{{1},{2,3}}
=> {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => ? = 1
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ? = 2
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ? = 0
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => ? = 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ? = 1
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => ? = 0
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ? = 0
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ? = 1
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => ? = 1
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ? = 0
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ? = 0
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ? = 0
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => ? = 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => ? = 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ? = 1
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => ? = 2
{{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,4,5,3] => ? = 1
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
{{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => ? = 1
{{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => ? = 0
Description
The number of edges in the reduced word graph of a signed permutation.
The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [[1],[]]
=> ([],1)
=> ? = 0 + 2
{{1,2}}
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> ? = 0 + 2
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? = 0 + 2
{{1,2,3}}
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 2 = 0 + 2
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 2
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 2
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 1 + 2
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2 = 0 + 2
{{1,2,3,4}}
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 2
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 1 + 2
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 0 + 2
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 2
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 1 + 2
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 0 + 2
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 0 + 2
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 1 + 2
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 0 + 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 1 + 2
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 2 + 2
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 1 + 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 0 + 2
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [[1],[]]
=> ([],1)
=> ? = 0 + 3
{{1,2}}
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> ? = 0 + 3
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? = 0 + 3
{{1,2,3}}
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 3
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 0 + 3
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 1 + 3
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3 = 0 + 3
{{1,2,3,4}}
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 1 + 3
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 3
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 3
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 1 + 3
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 3
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 0 + 3
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 3
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 1 + 3
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 0 + 3
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 0 + 3
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 3
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 1 + 3
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 0 + 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 1 + 3
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 2 + 3
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 1 + 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 3
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 0 + 3
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000102
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000102: Semistandard tableaux ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000102: Semistandard tableaux ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
{{1}}
=> [1] => [[1]]
=> [[1]]
=> 0
{{1,2}}
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 0
{{1},{2}}
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
{{1,2,3}}
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ? = 0
{{1,2},{3}}
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 0
{{1,3},{2}}
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 0
{{1},{2,3}}
=> [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ? = 1
{{1},{2},{3}}
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
{{1,2,3,4}}
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ? = 1
{{1,2,3},{4}}
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ? = 2
{{1,2,4},{3}}
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ? = 0
{{1,2},{3,4}}
=> [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ? = 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 1
{{1,3,4},{2}}
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ? = 0
{{1,3},{2,4}}
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ? = 1
{{1,4},{2,3}}
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ? = 1
{{1},{2,3,4}}
=> [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ? = 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ? = 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ? = 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ? = 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ? = 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ? = 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ? = 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,3,3],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ? = 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ? = 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ? = 0
Description
The charge of a semistandard tableau.
Matching statistic: St001964
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
{{1}}
=> [1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
{{1,2}}
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1},{2}}
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1,2,3}}
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0
{{1,2},{3}}
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 2
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 0
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000181
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
{{1}}
=> [1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
{{1,2},{3}}
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 0 + 1
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 2 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 1 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001890
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
{{1}}
=> [1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
{{1,2},{3}}
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 0 + 1
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 2 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 1 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 1 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 1 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
Description
The maximum magnitude of the Möbius function of a poset.
The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
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