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Your data matches 83 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St001796
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 0
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 0
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 3
([],4)
=> 0
([(2,3)],4)
=> 0
([(1,3),(2,3)],4)
=> 0
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 0
([(0,3),(1,2),(2,3)],4)
=> 1
([(1,2),(1,3),(2,3)],4)
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
([],5)
=> 0
([(3,4)],5)
=> 0
([(2,4),(3,4)],5)
=> 0
([(1,4),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> 0
([(1,4),(2,3),(3,4)],5)
=> 0
([(0,1),(2,4),(3,4)],5)
=> 0
([(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 11
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 21
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 20
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 45
Description
The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1).
Matching statistic: St000781
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Values
([],1)
=> [1]
=> []
=> ? = 1 + 1
([],2)
=> [1,1]
=> [1]
=> 1 = 0 + 1
([(0,1)],2)
=> [2]
=> []
=> ? = 1 + 1
([],3)
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 3 + 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 4 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 4 + 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 3 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 3 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 5 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 9 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 5 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 11 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 21 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 4 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 20 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 12 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 45 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 75 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 125 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 3 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 2 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 3 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 4 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 3 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 4 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 5 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 4 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 6 + 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 3 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 9 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 2 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 5 + 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Values
([],1)
=> [1]
=> []
=> ? = 1 + 1
([],2)
=> [1,1]
=> [1]
=> 1 = 0 + 1
([(0,1)],2)
=> [2]
=> []
=> ? = 1 + 1
([],3)
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? = 3 + 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 4 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? = 4 + 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 3 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 3 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 4 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 3 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 5 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 9 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? = 5 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 11 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 21 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 4 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 4 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 20 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? = 12 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 45 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 75 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? = 125 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1 = 0 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 3 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 = 0 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 2 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 3 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 4 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 3 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 4 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 5 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? = 4 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 6 + 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1 = 0 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 3 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 9 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 = 0 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 2 + 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? = 5 + 1
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St000205
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? = 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? = 3
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [2]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 4
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 11
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 21
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 20
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 45
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 75
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 125
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [2,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 5
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 6
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [4]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 9
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 5
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)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> 0
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? = 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ?
=> ? = 3
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [2]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ?
=> ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ?
=> ? = 4
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [2]
=> 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ?
=> ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 11
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 21
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 20
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ?
=> ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 45
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 75
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ?
=> ? = 125
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [2,1]
=> 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [1,1]
=> 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 5
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ?
=> ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 6
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [4]
=> 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [2]
=> 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 9
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ?
=> ? = 5
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
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Values
([],1)
=> [1]
=> []
=> ? => ? = 1
([],2)
=> [1,1]
=> [1]
=> 1 => 0
([(0,1)],2)
=> [2]
=> []
=> ? => ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 0
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 3
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? => ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 4
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? => ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 11
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 21
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 20
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 45
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 75
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 125
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 5
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 6
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 9
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 5
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
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Values
([],1)
=> [1]
=> []
=> ? => ? = 1
([],2)
=> [1,1]
=> [1]
=> 1 => 0
([(0,1)],2)
=> [2]
=> []
=> ? => ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 0
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 3
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? => ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 4
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? => ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 11
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 21
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 20
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 45
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 75
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 125
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 5
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 6
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 9
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 5
Description
The number of descents of a binary word.
Matching statistic: St000293
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Values
([],1)
=> [1]
=> []
=> ? => ? = 1
([],2)
=> [1,1]
=> [1]
=> 1 => 0
([(0,1)],2)
=> [2]
=> []
=> ? => ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 0
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 3
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? => ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 4
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? => ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 11
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 21
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 20
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 45
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 75
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 125
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 5
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 6
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 9
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 5
Description
The number of inversions of a binary word.
Matching statistic: St000296
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000296: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Values
([],1)
=> [1]
=> []
=> => ? = 1
([],2)
=> [1,1]
=> [1]
=> 10 => 0
([(0,1)],2)
=> [2]
=> []
=> => ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> 110 => 0
([(1,2)],3)
=> [2,1]
=> [1]
=> 10 => 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> => ? = 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> => ? = 3
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 110 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 10 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> => ? = 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 100 => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> => ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 10 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> => ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> => ? = 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> => ? = 4
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1010 => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 100 => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> => ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> => ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 100 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> => ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 11
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 21
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 20
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> => ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 45
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 75
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> => ? = 125
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1010 => 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 100 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> => ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1000 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> => ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> => ? = 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 5
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> => ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 6
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1100 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 100 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1010 => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 100 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 9
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 10 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> => ? = 5
Description
The length of the symmetric border of a binary word.
The symmetric border of a word is the longest word which is a prefix and its reverse is a suffix.
The statistic value is equal to the length of the word if and only if the word is [[https://en.wikipedia.org/wiki/Palindrome|palindromic]].
Matching statistic: St000347
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000347: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000347: Binary words ⟶ ℤResult quality: 0% ●values known / values provided: 21%●distinct values known / distinct values provided: 0%
Values
([],1)
=> [1]
=> []
=> ? => ? = 1
([],2)
=> [1,1]
=> [1]
=> 1 => 0
([(0,1)],2)
=> [2]
=> []
=> ? => ? = 1
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 0
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 0
([(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> []
=> ? => ? = 3
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 0
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(2,3)],4)
=> [4]
=> []
=> ? => ? = 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> []
=> ? => ? = 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> []
=> ? => ? = 4
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 0
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 5
([(0,4),(1,3),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 9
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> []
=> ? => ? = 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 11
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 21
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 20
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [5]
=> []
=> ? => ? = 12
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 45
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 75
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> []
=> ? => ? = 125
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 0
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 0
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 5
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [6]
=> []
=> ? => ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 6
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 0
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 9
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 0
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> [6]
=> []
=> ? => ? = 5
Description
The inversion sum of a binary word.
A pair $a < b$ is an inversion of a binary word $w = w_1 \cdots w_n$ if $w_a = 1 > 0 = w_b$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$.
The following 73 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. St000183The side length of the Durfee square of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000847The number of standard Young tableaux whose descent set is the binary word. St000913The number of ways to refine the partition into singletons. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001722The number of minimal chains with small intervals between a binary word and the top element. St000630The length of the shortest palindromic decomposition of a binary word. St001471The magnitude of a Dyck path. 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. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000929The constant term of the character polynomial of an integer partition. St000944The 3-degree of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees.
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