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Your data matches 230 different statistics following compositions of up to 3 maps.
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Matching statistic: St000228
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00275: Graphs āto edge-partition of connected componentsā¶ Integer partitions
St000228: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000228: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 1
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [3]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [2,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [4]
=> 4
([],6)
=> []
=> 0
([(4,5)],6)
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [2]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [3]
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> 5
([(2,5),(3,4)],6)
=> [1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [3]
=> 3
([(1,2),(3,5),(4,5)],6)
=> [2,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [4]
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> [3,1]
=> 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [5]
=> 5
([(0,5),(1,5),(2,4),(3,4)],6)
=> [2,2]
=> 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [5]
=> 5
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [5]
=> 5
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [4]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [3,1]
=> 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [5]
=> 5
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [5]
=> 5
([],7)
=> []
=> 0
([(5,6)],7)
=> [1]
=> 1
([(4,6),(5,6)],7)
=> [2]
=> 2
([(3,6),(4,6),(5,6)],7)
=> [3]
=> 3
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000384
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs āto edge-partition of biconnected componentsā¶ Integer partitions
St000384: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000384: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 1
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4
([],6)
=> []
=> 0
([(4,5)],6)
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(2,5),(3,4)],6)
=> [1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> 5
([],7)
=> []
=> 0
([(5,6)],7)
=> [1]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> 2
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 3
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000459
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00276: Graphs āto edge-partition of biconnected componentsā¶ Integer partitions
St000459: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000459: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 1
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4
([],6)
=> []
=> 0
([(4,5)],6)
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(2,5),(3,4)],6)
=> [1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> 5
([],7)
=> []
=> 0
([(5,6)],7)
=> [1]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> 2
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 3
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000784
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs āto edge-partition of biconnected componentsā¶ Integer partitions
St000784: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000784: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 0
([],2)
=> []
=> 0
([(0,1)],2)
=> [1]
=> 1
([],3)
=> []
=> 0
([(1,2)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 3
([],4)
=> []
=> 0
([(2,3)],4)
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 3
([],5)
=> []
=> 0
([(3,4)],5)
=> [1]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4
([],6)
=> []
=> 0
([(4,5)],6)
=> [1]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(2,5),(3,4)],6)
=> [1,1]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 5
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> 5
([],7)
=> []
=> 0
([(5,6)],7)
=> [1]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> 2
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 3
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
Matching statistic: St001622
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 0
([],2)
=> ([],1)
=> 0
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([],3)
=> ([],1)
=> 0
([(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([],4)
=> ([],1)
=> 0
([(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([],5)
=> ([],1)
=> 0
([(3,4)],5)
=> ([(0,1)],2)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([],6)
=> ([],1)
=> 0
([(4,5)],6)
=> ([(0,1)],2)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 5
([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 5
([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 5
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 5
([(0,5),(1,4),(2,3)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 5
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 5
([],7)
=> ([],1)
=> 0
([(5,6)],7)
=> ([(0,1)],2)
=> 1
([(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 3
Description
The number of join-irreducible elements of a lattice.
An element $j$ of a lattice $L$ is '''join irreducible''' if it is not the least element and if $j=x\vee y$, then $j\in\{x,y\}$ for all $x,y\in L$.
Matching statistic: St000063
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs āto edge-partition of biconnected componentsā¶ Integer partitions
St000063: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000063: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 1 = 0 + 1
([],2)
=> []
=> 1 = 0 + 1
([(0,1)],2)
=> [1]
=> 2 = 1 + 1
([],3)
=> []
=> 1 = 0 + 1
([(1,2)],3)
=> [1]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4 = 3 + 1
([],4)
=> []
=> 1 = 0 + 1
([(2,3)],4)
=> [1]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
([],5)
=> []
=> 1 = 0 + 1
([(3,4)],5)
=> [1]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([],6)
=> []
=> 1 = 0 + 1
([(4,5)],6)
=> [1]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4 = 3 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([],7)
=> []
=> 1 = 0 + 1
([(5,6)],7)
=> [1]
=> 2 = 1 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> 3 = 2 + 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 4 = 3 + 1
Description
The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
Matching statistic: St000108
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs āto edge-partition of biconnected componentsā¶ Integer partitions
St000108: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000108: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 1 = 0 + 1
([],2)
=> []
=> 1 = 0 + 1
([(0,1)],2)
=> [1]
=> 2 = 1 + 1
([],3)
=> []
=> 1 = 0 + 1
([(1,2)],3)
=> [1]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4 = 3 + 1
([],4)
=> []
=> 1 = 0 + 1
([(2,3)],4)
=> [1]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
([],5)
=> []
=> 1 = 0 + 1
([(3,4)],5)
=> [1]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([],6)
=> []
=> 1 = 0 + 1
([(4,5)],6)
=> [1]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4 = 3 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([],7)
=> []
=> 1 = 0 + 1
([(5,6)],7)
=> [1]
=> 2 = 1 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> 3 = 2 + 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 4 = 3 + 1
Description
The number of partitions contained in the given partition.
Matching statistic: St000532
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs āto edge-partition of biconnected componentsā¶ Integer partitions
St000532: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000532: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 1 = 0 + 1
([],2)
=> []
=> 1 = 0 + 1
([(0,1)],2)
=> [1]
=> 2 = 1 + 1
([],3)
=> []
=> 1 = 0 + 1
([(1,2)],3)
=> [1]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4 = 3 + 1
([],4)
=> []
=> 1 = 0 + 1
([(2,3)],4)
=> [1]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
([],5)
=> []
=> 1 = 0 + 1
([(3,4)],5)
=> [1]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([],6)
=> []
=> 1 = 0 + 1
([(4,5)],6)
=> [1]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4 = 3 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([],7)
=> []
=> 1 = 0 + 1
([(5,6)],7)
=> [1]
=> 2 = 1 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> 3 = 2 + 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 4 = 3 + 1
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St001400
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs āto edge-partition of biconnected componentsā¶ Integer partitions
St001400: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St001400: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> 1 = 0 + 1
([],2)
=> []
=> 1 = 0 + 1
([(0,1)],2)
=> [1]
=> 2 = 1 + 1
([],3)
=> []
=> 1 = 0 + 1
([(1,2)],3)
=> [1]
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> [1,1]
=> 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 4 = 3 + 1
([],4)
=> []
=> 1 = 0 + 1
([(2,3)],4)
=> [1]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [1,1]
=> 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
([(0,3),(1,2)],4)
=> [1,1]
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 4 = 3 + 1
([],5)
=> []
=> 1 = 0 + 1
([(3,4)],5)
=> [1]
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> [1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([(1,4),(2,3)],5)
=> [1,1]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 5 = 4 + 1
([],6)
=> []
=> 1 = 0 + 1
([(4,5)],6)
=> [1]
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> [1,1]
=> 3 = 2 + 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(2,5),(3,4)],6)
=> [1,1]
=> 3 = 2 + 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 4 = 3 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> 4 = 3 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 5 = 4 + 1
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> 6 = 5 + 1
([],7)
=> []
=> 1 = 0 + 1
([(5,6)],7)
=> [1]
=> 2 = 1 + 1
([(4,6),(5,6)],7)
=> [1,1]
=> 3 = 2 + 1
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> 4 = 3 + 1
Description
The total number of Littlewood-Richardson tableaux of given shape.
This is the multiplicity of the Schur function $s_\lambda$ in $\sum_{\mu, \nu} s_\mu s_\nu$, where the sum is over all partitions $\mu$ and $\nu$.
Matching statistic: St000013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00276: Graphs āto edge-partition of biconnected componentsā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St000013: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St000013: Dyck paths ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
([],1)
=> []
=> []
=> 0
([],2)
=> []
=> []
=> 0
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> 1
([],3)
=> []
=> []
=> 0
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([],4)
=> []
=> []
=> 0
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([],5)
=> []
=> []
=> 0
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
([],6)
=> []
=> []
=> 0
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,5),(1,5),(2,4),(3,4)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,5),(1,4),(2,3)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,1),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
([],7)
=> []
=> []
=> 0
([(5,6)],7)
=> [1]
=> [1,0,1,0]
=> 1
([(4,6),(5,6)],7)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
([(3,6),(4,6),(5,6)],7)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
The following 220 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000171The degree of the graph. St000185The weighted size of a partition. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001034The area of the parallelogram polyomino associated with the Dyck path. St001746The coalition number of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000011The number of touch points (or returns) of a Dyck path. St000012The area of a Dyck path. St000025The number of initial rises of a Dyck path. St000676The number of odd rises of a Dyck path. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000984The number of boxes below precisely one peak. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000169The cocharge of a standard tableau. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000293The number of inversions of a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000444The length of the maximal rise of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St000393The number of strictly increasing runs in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000395The sum of the heights of the peaks of a Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000921The number of internal inversions of a binary word. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001267The length of the Lyndon factorization of the binary word. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001437The flex of a binary word. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001658The total number of rook placements on a Ferrers board. St001955The number of natural descents for set-valued two row standard Young tableaux. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000144The pyramid weight of the Dyck path. St000290The major index of a binary word. St000678The number of up steps after the last double rise of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000983The length of the longest alternating subword. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001203We associate to 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 Dyck path as follows:
St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001485The modular major index of a binary word. St001614The cyclic permutation representation number of a skew partition. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St000306The bounce count of a Dyck path. St000369The dinv deficit of a Dyck path. St000439The position of the first down step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000674The number of hills of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000874The position of the last double rise in a Dyck path. St000876The number of factors in the Catalan decomposition of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001584The area statistic between a Dyck path and its bounce path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001688The sum of the squares of the heights of the peaks of a Dyck path. St001725The harmonious chromatic number of a graph. St000681The Grundy value of Chomp on Ferrers diagrams. St000744The length of the path to the largest entry in a standard Young tableau. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001959The product of the heights of the peaks of a Dyck path. St000014The number of parking functions supported by a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001480The number of simple summands of the module J^2/J^3. St001933The largest multiplicity of a part in an integer partition. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001176The size of a partition minus its first part. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000015The number of peaks of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St000331The number of upper interactions of a Dyck path. St000667The greatest common divisor of the parts of the partition. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001389The number of partitions of the same length below the given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001571The Cartan determinant of the integer partition. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000081The number of edges of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000993The multiplicity of the largest part of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000454The largest eigenvalue of a graph if it is integral. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001311The cyclomatic number of a graph. St001341The number of edges in the center of a graph. St000448The number of pairs of vertices of a graph with distance 2. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001697The shifted natural comajor index of a standard Young tableau. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000450The number of edges minus the number of vertices plus 2 of a graph. St001621The number of atoms of a lattice. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000456The monochromatic index of a connected graph. St001645The pebbling number of a connected graph. St000910The number of maximal chains of minimal length in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2.
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