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Your data matches 17 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St001492
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 5 = 4 + 1
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St000213
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 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: St001211
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 5 = 4 + 1
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St000105
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => {{1}}
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 4
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000507
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [[1]]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [[1,2]]
=> 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [[1,2],[3]]
=> 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [[1,2,3],[4]]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[1,3],[2,4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[1,2,4],[3]]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[1,2],[3,4]]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 3
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [[1,2,3,4],[5]]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [[1,3,4],[2,5]]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,2,3,5],[4]]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [[1,2,4],[3,5]]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [[1,2,3],[4,5]]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [[1,3,4],[2,5]]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,3,4],[2,5]]
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [[1,3,5],[2,4]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [[1,2,4,5],[3]]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[1,3,5],[2,4]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [[1,2,3],[4,5]]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [[1,2,3],[4,5]]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [[1,2,4],[3,5]]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [[1,2,4],[3,5]]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[1,3,5],[2,4]]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[1,2,5],[3,4]]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 4
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000155
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [3,2,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [2,3,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [4,3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [4,3,1,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [1,4,3,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [3,4,2,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [2,4,1,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [5,3,4,1,2] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,2,1,5,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [4,3,1,5,2] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [1,3,5,4,2] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,3,5,2,1] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [3,2,5,1,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [4,3,5,1,2] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [2,5,4,1,3] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [3,1,4,5,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [3,5,4,2,1] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [1,4,3,5,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [1,5,4,3,2] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [5,4,3,2,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,4,2,1,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [3,5,4,1,2] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,4,3,1,2] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,1] => 3 = 4 - 1
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St000245
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,4,1,5,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [2,3,5,1,4] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [3,4,5,1,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [2,4,5,1,3] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [2,4,1,3,5] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 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)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 2 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [2,3,1] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => [3,2,4,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [2,3,5,1,4] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 3 = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => 2 = 3 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2 = 3 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [2,4,5,1,3] => 3 = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [2,4,3,5,1] => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [2,5,1,3,4] => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,5,2] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2 = 3 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 3 = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => [3,4,5,1,2] => 3 = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => [3,4,1,2,5] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => [3,2,4,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => [3,4,2,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => [3,2,4,1,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => [3,5,1,2,4] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => [3,2,5,1,4] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 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: St000702
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => 4
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 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: St000925
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 3
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,1,3,4,2] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,1,3,2,5] => {{1,4},{2},{3},{5}}
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => {{1,4},{2,3},{5}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 4
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000083The number of left oriented leafs of a binary tree except the first one. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000354The number of recoils of a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001863The number of weak excedances of a signed permutation. St001626The number of maximal proper sublattices of a lattice.
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