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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000213
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 1
{{1,2}}
=> [2,1] => [1,2] => 2
{{1},{2}}
=> [1,2] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 3
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => 2
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => 2
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => 2
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => 2
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => 3
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => 3
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => 4
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => 4
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => 4
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => 4
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => 4
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => 4
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => 4
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St001863
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 1
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 2
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 3
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [1,3,2] => 2
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [2,1,3] => 2
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [3,2,1] => 2
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [2,3,1] => 2
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 3
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [2,3,1,4] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 3
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 4
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 4
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 4
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => 4
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => 4
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => 4
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 4
Description
The number of weak excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 1 = 2 - 1
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 2 = 3 - 1
{{1,2},{3}}
=> [2,1,3] => [3,2,1] => [2,3,1] => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1 = 2 - 1
{{1},{2,3}}
=> [1,3,2] => [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [3,1,2] => 1 = 2 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 2 = 3 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,2,4,1] => [2,4,1,3] => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 2 = 3 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,2,3,4,1] => [2,3,4,5,1] => 3 = 4 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 4 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,2,3,1,5] => [2,3,4,1,5] => 3 = 4 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [4,2,3,5,1] => [2,3,5,1,4] => 3 = 4 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 3 = 4 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,3,4] => [2,5,4,3,1] => 1 = 2 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,2,4,3,1] => [2,4,3,5,1] => 2 = 3 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 3 = 4 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,2,1,4,5] => [2,3,1,4,5] => 3 = 4 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [3,2,5,4,1] => [2,4,5,1,3] => 3 = 4 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 4 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 3 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [3,2,4,1,5] => [2,4,1,3,5] => 3 = 4 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => 3 = 4 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0 = 1 - 1
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 1 = 2 - 1
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 0 = 1 - 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 2 = 3 - 1
{{1,2},{3}}
=> [2,1,3] => [3,2,1] => [2,3,1] => 1 = 2 - 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1 = 2 - 1
{{1},{2,3}}
=> [1,3,2] => [2,1,3] => [2,1,3] => 1 = 2 - 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [3,1,2] => 1 = 2 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 2 = 3 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,2,4,1] => [2,4,1,3] => 2 = 3 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 0 = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 2 = 3 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2 = 3 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 2 = 3 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4 = 5 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,2,3,4,1] => [2,3,4,5,1] => 3 = 4 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3 = 4 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,2,3,1,5] => [2,3,4,1,5] => 3 = 4 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [4,2,3,5,1] => [2,3,5,1,4] => 3 = 4 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 3 = 4 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,3,4] => [2,5,4,3,1] => 1 = 2 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,2,4,3,1] => [2,4,3,5,1] => 2 = 3 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 3 = 4 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,2,1,4,5] => [2,3,1,4,5] => 3 = 4 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [3,2,5,4,1] => [2,4,5,1,3] => 3 = 4 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,3,4] => 3 = 4 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 3 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [3,2,4,1,5] => [2,4,1,3,5] => 3 = 4 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => 3 = 4 - 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St000250
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
St000250: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 1 + 1
{{1,2}}
=> 3 = 2 + 1
{{1},{2}}
=> 2 = 1 + 1
{{1,2,3}}
=> 4 = 3 + 1
{{1,2},{3}}
=> 3 = 2 + 1
{{1,3},{2}}
=> 3 = 2 + 1
{{1},{2,3}}
=> 3 = 2 + 1
{{1},{2},{3}}
=> 3 = 2 + 1
{{1,2,3,4}}
=> 5 = 4 + 1
{{1,2,3},{4}}
=> 4 = 3 + 1
{{1,2,4},{3}}
=> 4 = 3 + 1
{{1,2},{3,4}}
=> 4 = 3 + 1
{{1,2},{3},{4}}
=> 4 = 3 + 1
{{1,3,4},{2}}
=> 4 = 3 + 1
{{1,3},{2,4}}
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> 3 = 2 + 1
{{1,4},{2,3}}
=> 4 = 3 + 1
{{1},{2,3,4}}
=> 4 = 3 + 1
{{1},{2,3},{4}}
=> 4 = 3 + 1
{{1,4},{2},{3}}
=> 4 = 3 + 1
{{1},{2,4},{3}}
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> 4 = 3 + 1
{{1},{2},{3},{4}}
=> 4 = 3 + 1
{{1,2,3,4,5}}
=> 6 = 5 + 1
{{1,2,3,4},{5}}
=> 5 = 4 + 1
{{1,2,3,5},{4}}
=> 5 = 4 + 1
{{1,2,3},{4,5}}
=> 5 = 4 + 1
{{1,2,3},{4},{5}}
=> 5 = 4 + 1
{{1,2,4,5},{3}}
=> 5 = 4 + 1
{{1,2,4},{3,5}}
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> 4 = 3 + 1
{{1,2,5},{3,4}}
=> 5 = 4 + 1
{{1,2},{3,4,5}}
=> 5 = 4 + 1
{{1,2},{3,4},{5}}
=> 5 = 4 + 1
{{1,2,5},{3},{4}}
=> 5 = 4 + 1
{{1,2},{3,5},{4}}
=> 4 = 3 + 1
{{1,2},{3},{4,5}}
=> 5 = 4 + 1
{{1,2},{3},{4},{5}}
=> 5 = 4 + 1
Description
The number of blocks ([[St000105]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000702
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => ? = 1
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 2
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 3
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [1,3,2] => 2
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [2,1,3] => 2
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [3,2,1] => 2
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [3,1,2] => 2
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 3
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 3
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 3
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [3,1,2,4] => 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 4
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 4
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,3,4,5,2] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 4
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,5,3,2,4] => 4
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [1,4,2,3,5] => 4
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,5,2,4,3] => 4
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 4
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St001604
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 20%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 20%
Values
{{1}}
=> [1]
=> [1]
=> []
=> ? = 1 - 4
{{1,2}}
=> [2]
=> [1,1]
=> [1]
=> ? = 2 - 4
{{1},{2}}
=> [1,1]
=> [2]
=> []
=> ? = 1 - 4
{{1,2,3}}
=> [3]
=> [3]
=> []
=> ? = 3 - 4
{{1,2},{3}}
=> [2,1]
=> [1,1,1]
=> [1,1]
=> ? = 2 - 4
{{1,3},{2}}
=> [2,1]
=> [1,1,1]
=> [1,1]
=> ? = 2 - 4
{{1},{2,3}}
=> [2,1]
=> [1,1,1]
=> [1,1]
=> ? = 2 - 4
{{1},{2},{3}}
=> [1,1,1]
=> [2,1]
=> [1]
=> ? = 2 - 4
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 4 - 4
{{1,2,3},{4}}
=> [3,1]
=> [3,1]
=> [1]
=> ? = 3 - 4
{{1,2,4},{3}}
=> [3,1]
=> [3,1]
=> [1]
=> ? = 3 - 4
{{1,2},{3,4}}
=> [2,2]
=> [4]
=> []
=> ? = 3 - 4
{{1,2},{3},{4}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> ? = 3 - 4
{{1,3,4},{2}}
=> [3,1]
=> [3,1]
=> [1]
=> ? = 3 - 4
{{1,3},{2,4}}
=> [2,2]
=> [4]
=> []
=> ? = 1 - 4
{{1,3},{2},{4}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> ? = 2 - 4
{{1,4},{2,3}}
=> [2,2]
=> [4]
=> []
=> ? = 3 - 4
{{1},{2,3,4}}
=> [3,1]
=> [3,1]
=> [1]
=> ? = 3 - 4
{{1},{2,3},{4}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> ? = 3 - 4
{{1,4},{2},{3}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> ? = 3 - 4
{{1},{2,4},{3}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> ? = 2 - 4
{{1},{2},{3,4}}
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> ? = 3 - 4
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [2,2]
=> [2]
=> ? = 3 - 4
{{1,2,3,4,5}}
=> [5]
=> [5]
=> []
=> ? = 5 - 4
{{1,2,3,4},{5}}
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 4 - 4
{{1,2,3,5},{4}}
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 4 - 4
{{1,2,3},{4,5}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> ? = 4 - 4
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,2]
=> [2]
=> ? = 4 - 4
{{1,2,4,5},{3}}
=> [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 4 - 4
{{1,2,4},{3,5}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> ? = 2 - 4
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,2]
=> [2]
=> ? = 3 - 4
{{1,2,5},{3,4}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> ? = 4 - 4
{{1,2},{3,4,5}}
=> [3,2]
=> [3,1,1]
=> [1,1]
=> ? = 4 - 4
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [4,1]
=> [1]
=> ? = 4 - 4
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,2]
=> [2]
=> ? = 4 - 4
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [4,1]
=> [1]
=> ? = 3 - 4
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [4,1]
=> [1]
=> ? = 4 - 4
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0 = 4 - 4
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001880
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
{{1}}
=> [1] => ([],1)
=> ? = 1 + 1
{{1,2}}
=> [2,1] => ([],2)
=> ? = 2 + 1
{{1},{2}}
=> [1,2] => ([(0,1)],2)
=> ? = 1 + 1
{{1,2,3}}
=> [2,3,1] => ([(1,2)],3)
=> ? = 3 + 1
{{1,2},{3}}
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
{{1,3},{2}}
=> [3,2,1] => ([],3)
=> ? = 2 + 1
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ? = 2 + 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? = 4 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? = 3 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? = 3 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? = 3 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? = 3 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([],4)
=> ? = 3 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? = 3 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(2,3)],4)
=> ? = 3 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ? = 2 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? = 3 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ? = 5 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ? = 4 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? = 4 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 4 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 2 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 3 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 4 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? = 4 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ? = 4 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ? = 4 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ? = 3 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? = 4 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 + 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000264
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
{{1}}
=> [1] => [1] => ([],1)
=> ? = 1
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> ? = 2
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> ? = 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> ? = 3
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? = 2
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => ([],3)
=> ? = 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 2
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? = 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3
{{1,3},{2,4}}
=> [3,4,1,2] => [1,2,3,4] => ([],4)
=> ? = 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? = 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,2,3,4] => ([],4)
=> ? = 3
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 3
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ? = 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 4
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 4
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 4
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 4
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000736
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
St000736: Semistandard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000736: Semistandard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
{{1}}
=> [1] => [[1]]
=> [[1]]
=> 1
{{1,2}}
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 2
{{1},{2}}
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 1
{{1,2,3}}
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ? = 3
{{1,2},{3}}
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 2
{{1,3},{2}}
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 2
{{1},{2,3}}
=> [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ? = 2
{{1},{2},{3}}
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 2
{{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]]
=> ? = 4
{{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]]
=> ? = 3
{{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]]
=> ? = 3
{{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]]
=> ? = 3
{{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]]
=> ? = 3
{{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]]
=> ? = 3
{{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]]
=> ? = 1
{{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]]
=> ? = 2
{{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]]
=> ? = 3
{{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]]
=> ? = 3
{{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]]
=> ? = 3
{{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]]
=> ? = 3
{{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]]
=> ? = 2
{{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]]
=> ? = 3
{{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]]
=> ? = 3
{{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]]
=> ? = 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ? = 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ? = 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ? = 4
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ? = 4
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ? = 4
{{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]]
=> ? = 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ? = 3
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ? = 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ? = 4
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ? = 4
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ? = 4
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ? = 3
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ? = 4
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ? = 4
Description
The last entry in the first row of a semistandard tableau.
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