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Your data matches 58 different statistics following compositions of up to 3 maps.
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Matching statistic: St001268
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Values
([],1)
=> ([],1)
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 4
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 4
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 3
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 3
Description
The size of the largest ordinal summand in the poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.
Matching statistic: St000987
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Values
([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 3 = 4 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 3 = 4 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0 = 1 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> 2 = 3 - 1
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St000147
Values
([],1)
=> ([],1)
=> ([],1)
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St001389
Values
([],1)
=> ([],1)
=> ([],1)
=> [1]
=> 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000225
Values
([],1)
=> ([],1)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000319
Values
([],1)
=> ([],1)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Values
([],1)
=> ([],1)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001918
Values
([],1)
=> ([],1)
=> ([],1)
=> [1]
=> 0 = 1 - 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 0 = 1 - 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 3 = 4 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 2 = 3 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St000668
Values
([],1)
=> ([],1)
=> ([],1)
=> [1]
=> ? = 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
Values
([],1)
=> ([],1)
=> ([],1)
=> [1]
=> ? = 1
([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> [1,1]
=> 1
([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> [3,1,1]
=> 3
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> [1,1,1,1,1]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> [2,1,1,1,1]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 4
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> [1,1,1,1,1,1]
=> 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 3
Description
The product of the parts of an integer partition.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001128The exponens consonantiae of a partition. St000454The largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000909The number of maximal chains of maximal size in a poset. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001624The breadth of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001060The distinguishing index of a graph. St000632The jump number of the poset. St000100The number of linear extensions of a poset. St000307The number of rowmotion orbits of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001271The competition number of a graph. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St000264The girth of a graph, which is not a tree. St001964The interval resolution global dimension of a poset. St001875The number of simple modules with projective dimension at most 1. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001890The maximum magnitude of the Möbius function of a poset. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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