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Your data matches 483 different statistics following compositions of up to 3 maps.
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Matching statistic: St001176
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 0
([],2)
=> [1,1]
=> 1
([(0,1)],2)
=> [2]
=> 0
([],3)
=> [1,1,1]
=> 2
([(1,2)],3)
=> [2,1]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> 1
([],4)
=> [1,1,1,1]
=> 3
([(2,3)],4)
=> [2,1,1]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
([(1,2),(2,3)],4)
=> [3,1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 1
([],5)
=> [1,1,1,1,1]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000010
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1 = 0 + 1
([],2)
=> [1,1]
=> 2 = 1 + 1
([(0,1)],2)
=> [2]
=> 1 = 0 + 1
([],3)
=> [1,1,1]
=> 3 = 2 + 1
([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> [2,1]
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> [3]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
([(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 2 = 1 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
([(1,2),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
([],5)
=> [1,1,1,1,1]
=> 5 = 4 + 1
([(3,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> 2 = 1 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> 2 = 1 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St001494
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
Description
The Alon-Tarsi number of a graph.
Let $G$ be a graph with vertices $\{1,\dots,n\}$ and edge set $E$. Let $P_G=\prod_{i < j, (i,j)\in E} x_i-x_j$ be its graph polynomial. Then the Alon-Tarsi number is the smallest number $k$ such that $P_G$ contains a monomial with exponents strictly less than $k$.
Matching statistic: St001581
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2 = 1 + 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> 2 = 1 + 1
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000012
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [1,0]
=> 0
([],2)
=> [1,1]
=> [1,1,0,0]
=> 1
([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 0
([],3)
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
([(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000148
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> []
=> 0
([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> [2]
=> []
=> 0
([],3)
=> [1,1,1]
=> [1,1]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> 1
Description
The number of odd parts of a partition.
Matching statistic: St000157
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> [[1]]
=> 0
([],2)
=> [1,1]
=> [[1],[2]]
=> 1
([(0,1)],2)
=> [2]
=> [[1,2]]
=> 0
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1
([(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000160
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> []
=> 0
([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> [2]
=> []
=> 0
([],3)
=> [1,1,1]
=> [1,1]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000228
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> []
=> 0
([],2)
=> [1,1]
=> [1]
=> 1
([(0,1)],2)
=> [2]
=> []
=> 0
([],3)
=> [1,1,1]
=> [1,1]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> 0
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000272
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 0
([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,1)],2)
=> ([],2)
=> ([],1)
=> 0
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> 0
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> 0
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 1
Description
The treewidth of a graph.
A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
The following 473 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000362The size of a minimal vertex cover of a graph. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000536The pathwidth of a graph. St000548The number of different non-empty partial sums of an integer partition. St000632The jump number of the poset. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St001127The sum of the squares of the parts of a partition. St001777The number of weak descents in an integer composition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000527The width of the poset. St000532The total number of rook placements on a Ferrers board. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000013The height of a Dyck path. St000019The cardinality of the support of a permutation. St000041The number of nestings of a perfect matching. St000053The number of valleys of the Dyck path. St000141The maximum drop size of a permutation. St000214The number of adjacencies of a permutation. St000293The number of inversions of a binary word. St000377The dinv defect of an integer partition. St000392The length of the longest run of ones in a binary word. St000546The number of global descents of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000662The staircase size of the code of a permutation. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000921The number of internal inversions of a binary word. St000992The alternating sum of the parts of an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001090The number of pop-stack-sorts needed to sort a permutation. St001161The major index north count of a Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000026The position of the first return of a Dyck path. St000054The first entry of the permutation. St000058The order of a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000393The number of strictly increasing runs in a binary word. St000507The number of ascents of a standard tableau. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000722The number of different neighbourhoods in a graph. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000839The largest opener of a set partition. St000883The number of longest increasing subsequences of a permutation. St000982The length of the longest constant subword. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module 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:
St001267The length of the Lyndon factorization of the binary word. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001654The monophonic hull number of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001814The number of partitions interlacing the given partition. St000439The position of the first down step of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000806The semiperimeter of the associated bargraph. St000984The number of boxes below precisely one peak. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000209Maximum difference of elements in cycles. St000297The number of leading ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000446The disorder of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000925The number of topologically connected components of a set partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001808The box weight or horizontal decoration of a Dyck path. St000809The reduced reflection length of the permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000025The number of initial rises of a Dyck path. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001933The largest multiplicity of a part in an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000444The length of the maximal rise of a Dyck path. St000531The leading coefficient of the rook polynomial of an integer partition. St000627The exponent of a binary word. St000667The greatest common divisor of the parts of the partition. St000878The number of ones minus the number of zeros of a binary word. St000922The minimal number such that all substrings of this length are unique. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001415The length of the longest palindromic prefix of a binary word. 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. St001485The modular major index of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001884The number of borders of a binary word. St000145The Dyson rank of a partition. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000518The number of distinct subsequences in a binary word. 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. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001658The total number of rook placements on a Ferrers board. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001955The number of natural descents for set-valued two row standard Young tableaux. St000438The position of the last up step in a Dyck path. 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. 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. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. 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. 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. St000144The pyramid weight of the 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. St001018Sum of projective dimension of the indecomposable injective modules of 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. 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. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in 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 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. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001316The domatic number of a graph. St000069The number of maximal elements of a poset. St000068The number of minimal elements in a poset. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000006The dinv of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000339The maf index of a permutation. St000653The last descent of a permutation. St001497The position of the largest weak excedence of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000993The multiplicity of the largest part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000477The weight of a partition according to Alladi. St000770The major index of an integer partition when read from bottom to top. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000932The number of occurrences of the pattern UDU in a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000822The Hadwiger number of the graph. St001812The biclique partition number of a graph. St000741The Colin de Verdière graph invariant. St001644The dimension of a graph. St000071The number of maximal chains in a poset. St000161The sum of the sizes of the right subtrees of a binary tree. St000306The bounce count of a Dyck path. St000937The number of positive values of the symmetric group character corresponding to the partition. 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 $eA$. St000018The number of inversions of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000237The number of small exceedances. St000110The number of permutations less than or equal to a permutation in left weak order. St000246The number of non-inversions of a permutation. St000651The maximal size of a rise in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000926The clique-coclique number of a graph. St001110The 3-dynamic chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000005The bounce statistic of a Dyck path. St000008The major index of the composition. St000021The number of descents of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. 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. St001185The number of indecomposable injective modules of grade at least 2 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001340The cardinality of a minimal non-edge isolating set of a graph. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000015The number of peaks of a Dyck path. St000184The size of the centralizer of any permutation of given cycle type. St000240The number of indices that are not small excedances. St000258The burning number of a graph. St000273The domination number of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000325The width of the tree associated to a permutation. St000479The Ramsey number of a graph. St000482The (zero)-forcing number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000916The packing number of a graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000990The first ascent of a permutation. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001441The number of non-empty connected induced subgraphs of a graph. St001463The number of distinct columns in the nullspace of a graph. St001530The depth of a Dyck path. St001672The restrained domination number of a graph. St001675The number of parts equal to the part in the reversed composition. St001691The number of kings in a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000216The absolute length of a permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000082The number of elements smaller than a binary tree in Tamari order. St001346The number of parking functions that give the same permutation. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001834The number of non-isomorphic minors of a graph. St001706The number of closed sets in a graph. St001674The number of vertices of the largest induced star graph in the graph. St001323The independence gap of a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St001570The minimal number of edges to add to make a graph Hamiltonian. St000659The number of rises of length at least 2 of a Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. 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 $eA$. St001651The Frankl number of a lattice. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001488The number of corners of a skew partition. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001684The reduced word complexity of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St001568The smallest positive integer that does not appear twice in the partition. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000260The radius of a connected graph. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001060The distinguishing index of a graph. St000302The determinant of the distance matrix of a connected graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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