searching the database
Your data matches 65 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000929
(load all 64 compositions to match this statistic)
(load all 64 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000929: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> 0
{{1},{2}}
=> [1,1]
=> 1
{{1,2,3}}
=> [3]
=> 0
{{1,2},{3}}
=> [2,1]
=> 0
{{1,3},{2}}
=> [2,1]
=> 0
{{1},{2,3}}
=> [2,1]
=> 0
{{1},{2},{3}}
=> [1,1,1]
=> 1
{{1,2,3,4}}
=> [4]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> 0
{{1,2},{3},{4}}
=> [2,1,1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001125
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
Description
The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.
Matching statistic: St001139
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
{{1},{2}}
=> [1,2] => [1,2] => [1,0,1,0]
=> 0
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 0
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 0
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0
Description
The number of occurrences of hills of size 2 in a Dyck path.
A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.
Matching statistic: St001353
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001353: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001353: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1] => ([],1)
=> 1
{{1},{2}}
=> [1,1] => [2] => ([],2)
=> 0
{{1,2,3}}
=> [3] => [1] => ([],1)
=> 1
{{1,2},{3}}
=> [2,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,3},{2}}
=> [2,1] => [1,1] => ([(0,1)],2)
=> 0
{{1},{2,3}}
=> [1,2] => [1,1] => ([(0,1)],2)
=> 0
{{1},{2},{3}}
=> [1,1,1] => [3] => ([],3)
=> 0
{{1,2,3,4}}
=> [4] => [1] => ([],1)
=> 1
{{1,2,3},{4}}
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2},{3,4}}
=> [2,2] => [2] => ([],2)
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,3},{2,4}}
=> [2,2] => [2] => ([],2)
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,4},{2,3}}
=> [2,2] => [2] => ([],2)
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,1] => ([(0,1)],2)
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
{{1},{2},{3,4}}
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => ([],4)
=> 0
{{1,2,3,4,5}}
=> [5] => [1] => ([],1)
=> 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,2,5},{3,4}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
Description
The number of prime nodes in the modular decomposition of a graph.
Matching statistic: St001356
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001356: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001356: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1] => ([],1)
=> 1
{{1},{2}}
=> [1,1] => [2] => ([],2)
=> 0
{{1,2,3}}
=> [3] => [1] => ([],1)
=> 1
{{1,2},{3}}
=> [2,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,3},{2}}
=> [2,1] => [1,1] => ([(0,1)],2)
=> 0
{{1},{2,3}}
=> [1,2] => [1,1] => ([(0,1)],2)
=> 0
{{1},{2},{3}}
=> [1,1,1] => [3] => ([],3)
=> 0
{{1,2,3,4}}
=> [4] => [1] => ([],1)
=> 1
{{1,2,3},{4}}
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,4},{3}}
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2},{3,4}}
=> [2,2] => [2] => ([],2)
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,3,4},{2}}
=> [3,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,3},{2,4}}
=> [2,2] => [2] => ([],2)
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,4},{2,3}}
=> [2,2] => [2] => ([],2)
=> 0
{{1},{2,3,4}}
=> [1,3] => [1,1] => ([(0,1)],2)
=> 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
{{1},{2},{3,4}}
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => ([],4)
=> 0
{{1,2,3,4,5}}
=> [5] => [1] => ([],1)
=> 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,2,5},{3,4}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
Description
The number of vertices in prime modules of a graph.
Matching statistic: St001657
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => ([],2)
=> [1,1]
=> 0
{{1},{2}}
=> [1,1] => ([(0,1)],2)
=> [2]
=> 1
{{1,2,3}}
=> [3] => ([],3)
=> [1,1,1]
=> 0
{{1,2},{3}}
=> [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 0
{{1,3},{2}}
=> [2,1] => ([(0,2),(1,2)],3)
=> [3]
=> 0
{{1},{2,3}}
=> [1,2] => ([(1,2)],3)
=> [2,1]
=> 1
{{1},{2},{3}}
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 0
{{1,2,3,4}}
=> [4] => ([],4)
=> [1,1,1,1]
=> 0
{{1,2,3},{4}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
{{1,2,4},{3}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
{{1,2},{3,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 0
{{1,2},{3},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
{{1,3,4},{2}}
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> [4]
=> 0
{{1,3},{2,4}}
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 0
{{1,3},{2},{4}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
{{1,4},{2,3}}
=> [2,2] => ([(1,3),(2,3)],4)
=> [3,1]
=> 0
{{1},{2,3,4}}
=> [1,3] => ([(2,3)],4)
=> [2,1,1]
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
{{1,4},{2},{3}}
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
{{1},{2,4},{3}}
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
{{1},{2},{3,4}}
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
{{1,2,3,4,5}}
=> [5] => ([],5)
=> [1,1,1,1,1]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,2,3},{4,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,2,4},{3,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,2,5},{3,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
{{1,2},{3,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,3,4},{2,5}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,3,5},{2,4}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
{{1,3},{2,4,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
{{1,4,5},{2,3}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 0
{{1,4},{2,3,5}}
=> [2,3] => ([(2,4),(3,4)],5)
=> [3,1,1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St000205
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> []
=> ?
=> ? = 1
{{1},{2}}
=> [1,1]
=> [1]
=> [1]
=> 0
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? = 1
{{1,2},{3}}
=> [2,1]
=> [1]
=> [1]
=> 0
{{1,3},{2}}
=> [2,1]
=> [1]
=> [1]
=> 0
{{1},{2,3}}
=> [2,1]
=> [1]
=> [1]
=> 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? = 1
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> [1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> [1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> [1,1]
=> 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> [1]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> [1,1]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> [1,1]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> [1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? = 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> [1,1]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ?
=> ? = 1
{{1,2,3,4,5,6,7}}
=> [7]
=> []
=> ?
=> ? = 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2]
=> []
=> ?
=> ? = 1
{{1},{2}}
=> [1,1]
=> [1]
=> [1]
=> 0
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? = 1
{{1,2},{3}}
=> [2,1]
=> [1]
=> [1]
=> 0
{{1,3},{2}}
=> [2,1]
=> [1]
=> [1]
=> 0
{{1},{2,3}}
=> [2,1]
=> [1]
=> [1]
=> 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? = 1
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> [1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> [1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> [2]
=> 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> [1]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> [2]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> [2]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> [1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? = 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> [2]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> [1]
=> 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ?
=> ? = 1
{{1,2,3,4,5,6,7}}
=> [7]
=> []
=> ?
=> ? = 1
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000290
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2]
=> []
=> ? => ? = 1
{{1},{2}}
=> [1,1]
=> [1]
=> 1 => 0
{{1,2,3}}
=> [3]
=> []
=> ? => ? = 1
{{1,2},{3}}
=> [2,1]
=> [1]
=> 1 => 0
{{1,3},{2}}
=> [2,1]
=> [1]
=> 1 => 0
{{1},{2,3}}
=> [2,1]
=> [1]
=> 1 => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> 11 => 0
{{1,2,3,4}}
=> [4]
=> []
=> ? => ? = 1
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> 1 => 0
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> 1 => 0
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> 0 => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> 1 => 0
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> 0 => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> 0 => 0
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> 1 => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ? => ? = 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> 1 => 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> 1 => 0
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> 1 => 0
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> 1 => 0
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> 0 => 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> 1 => 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ? => ? = 1
{{1,2,3,4,5,6,7}}
=> [7]
=> []
=> ? => ? = 1
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000291
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2]
=> []
=> ? => ? = 1
{{1},{2}}
=> [1,1]
=> [1]
=> 1 => 0
{{1,2,3}}
=> [3]
=> []
=> ? => ? = 1
{{1,2},{3}}
=> [2,1]
=> [1]
=> 1 => 0
{{1,3},{2}}
=> [2,1]
=> [1]
=> 1 => 0
{{1},{2,3}}
=> [2,1]
=> [1]
=> 1 => 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> 11 => 0
{{1,2,3,4}}
=> [4]
=> []
=> ? => ? = 1
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> 1 => 0
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> 1 => 0
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> 0 => 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> 1 => 0
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> 0 => 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> 0 => 0
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> 1 => 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> 11 => 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
{{1,2,3,4,5}}
=> [5]
=> []
=> ? => ? = 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> 1 => 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> 1 => 0
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> 1 => 0
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> 1 => 0
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> 0 => 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> 0 => 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> 1 => 0
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> 11 => 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> 01 => 0
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ? => ? = 1
{{1,2,3,4,5,6,7}}
=> [7]
=> []
=> ? => ? = 1
Description
The number of descents of a binary word.
The following 55 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001175The size of a partition minus the hook length of the base cell. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000990The first ascent of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000546The number of global descents of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000234The number of global ascents of a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000315The number of isolated vertices of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001691The number of kings in a graph. St000056The decomposition (or block) number of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St000153The number of adjacent cycles of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001803The maximal overlap of the cylindrical tableau associated with a tableau.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!