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Your data matches 162 different statistics following compositions of up to 3 maps.
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Matching statistic: St000907
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Values
([],1)
=> 1
([],2)
=> 1
([(0,1)],2)
=> 2
([],3)
=> 1
([(1,2)],3)
=> 2
([(0,1),(0,2)],3)
=> 1
([(0,2),(2,1)],3)
=> 3
([(0,2),(1,2)],3)
=> 1
([],4)
=> 1
([(2,3)],4)
=> 2
([(1,2),(1,3)],4)
=> 1
([(0,1),(0,2),(0,3)],4)
=> 1
([(0,2),(0,3),(3,1)],4)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,2),(2,3)],4)
=> 3
([(0,3),(3,1),(3,2)],4)
=> 2
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(3,2)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 4
([(0,3),(1,2),(1,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> 4
([(0,3),(1,2),(2,3)],4)
=> 1
([],5)
=> 1
([(3,4)],5)
=> 2
([(2,3),(2,4)],5)
=> 1
([(1,2),(1,3),(1,4)],5)
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(4,2)],5)
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(2,3),(3,4)],5)
=> 3
([(1,4),(4,2),(4,3)],5)
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
([(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(4,3)],5)
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000160
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> [1]
=> 1
([],2)
=> ([(0,1)],2)
=> [2]
=> 1
([(0,1)],2)
=> ([],2)
=> [1,1]
=> 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2,2]
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [2,1]
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> [1,1,1]
=> 3
([(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]
=> 1
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,3]
=> 2
([(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)
=> [3,1]
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> [2,2,2]
=> 3
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [2,1,1]
=> 2
([(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]
=> 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2,2,2]
=> 4
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2,2]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> [1,1,1,1]
=> 4
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2,2,1]
=> 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,4]
=> 2
([(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)
=> [4,2]
=> 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,1]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,2]
=> 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,1]
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [2,1,1,1]
=> 3
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,2,2,1]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> [2,2,2,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 1
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,3,3]
=> 3
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,2,2]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(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)
=> [3,2,2]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> [2,2,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,2]
=> 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,3,2,2]
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,2,2,2]
=> 3
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: St001491
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(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> ? => ? = 1
([],2)
=> [1,1]
=> [1]
=> 1 => 1
([(0,1)],2)
=> [2]
=> []
=> ? => ? = 2
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 2
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> 1 => 1
([(0,2),(2,1)],3)
=> [3]
=> []
=> ? => ? = 3
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> 1 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 1
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 1
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> 1 => 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 1 => 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => ? ∊ {1,3,4,4}
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 0 => ? ∊ {1,3,4,4}
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 0 => ? ∊ {1,3,4,4}
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? => ? ∊ {1,3,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 3
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => ? ∊ {1,1,1,1,1,3,3,3,4,4,4,4,5,5,5,6}
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St000939
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> ? ∊ {1,2}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ? ∊ {1,1,1,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
([],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]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 5
([(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]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> 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
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> 1
([(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)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000993
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> ? ∊ {1,2}
([(0,1)],2)
=> [2]
=> []
=> ? ∊ {1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ? ∊ {1,1,1,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
([],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]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([],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]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> 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
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> 1
([(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)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000815
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> ?
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2}
([(0,1)],2)
=> [2]
=> []
=> ?
=> ? ∊ {1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ?
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,5,5,5,6}
Description
The number of semistandard Young tableaux of partition weight of given shape.
The weight of a semistandard Young tableaux is the sequence $(m_1, m_2,\dots)$, where $m_i$ is the number of occurrences of the number $i$ in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence.
Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.
Matching statistic: St000937
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 71%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [1]
=> []
=> []
=> ? = 1
([],2)
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2}
([(0,1)],2)
=> [2]
=> []
=> []
=> ? ∊ {1,2}
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,3}
([(0,1),(0,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,3}
([(0,2),(2,1)],3)
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,3}
([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,3}
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(3,1),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,3,4,4}
([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,3,4,4}
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 5
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> []
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,6}
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000771
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? = 2
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {2,3}
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {2,3}
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {2,2,3,3,4,4}
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ? ∊ {2,2,3,3,4,4}
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,4,4}
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {2,2,3,3,4,4}
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {2,2,3,3,4,4}
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {2,2,3,3,4,4}
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(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
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(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
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(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
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 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)
=> ([(0,2),(1,2)],3)
=> 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1
([(1,4),(3,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,5,5,5,6}
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([(0,1)],2)
=> ([],2)
=> ([],2)
=> ? = 2
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],3)
=> ? ∊ {1,3}
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? ∊ {1,3}
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 1
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([],1)
=> 1
([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],4)
=> ? ∊ {1,2,2,3,4,4}
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ? ∊ {1,2,2,3,4,4}
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,2,2,3,4,4}
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,2,2,3,4,4}
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([],2)
=> ? ∊ {1,2,2,3,4,4}
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? ∊ {1,2,2,3,4,4}
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([],1)
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],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
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],5)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(4,2)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([],3)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([],2)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([],1)
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(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
([(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)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(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)
=> 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([],1)
=> 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
([(1,4),(3,2),(4,3)],5)
=> ([(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)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
([(0,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,2,3,3,3,3,3,3,3,3,4,4,4,4,5,5,5,6}
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001571
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 67%
Values
([],1)
=> [2]
=> []
=> []
=> ? = 1
([],2)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(0,1)],2)
=> [3]
=> []
=> []
=> ? = 2
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(1,2)],3)
=> [6]
=> []
=> []
=> ? ∊ {1,2}
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(2,1)],3)
=> [4]
=> []
=> []
=> ? ∊ {1,2}
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? ∊ {1,2,4,4}
([(2,3)],4)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> []
=> ? ∊ {1,2,4,4}
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> [1,1,1,1]
=> 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> [2,2,2]
=> 2
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> [1,1,1]
=> 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> []
=> ? ∊ {1,2,4,4}
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> []
=> ? ∊ {1,2,4,4}
([],5)
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [15,15]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(3,4)],5)
=> [6,6,6,6]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(2,3),(2,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> [5,5,1,1,1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,1),(0,2),(0,3),(0,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(1,3),(1,4),(4,2)],5)
=> [14]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> [2,2,2]
=> 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(2,4),(3,4)],5)
=> [6,6,2,2,2,2]
=> [6,2,2,2,2]
=> [5,5,1,1,1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 3
([(0,4),(1,4),(2,4),(3,4)],5)
=> [3,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [7,7]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> [2,2,1]
=> 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> [3,3,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> [2,2,2,2,2,2]
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [3]
=> [1,1,1]
=> 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 4
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2]
=> [1,1]
=> 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,4),(1,2),(1,4),(4,3)],5)
=> [10]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [2]
=> [1,1]
=> 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [3,2]
=> [2,2,1]
=> 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [10]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(1,4),(3,2),(4,3)],5)
=> [10]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(1,4),(2,3),(3,4)],5)
=> [14]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,3),(1,4),(4,2)],5)
=> [12]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,4),(3,2),(4,1),(4,3)],5)
=> [8]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,4),(1,2),(2,3),(2,4)],5)
=> [10]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,4,4,5,5,5,6}
Description
The Cartan determinant of the integer partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
The following 152 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000454The largest eigenvalue of a graph if it is integral. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000189The number of elements in the poset. St000528The height of a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001118The acyclic chromatic index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000460The hook length of the last cell along the main diagonal of an integer partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer 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. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001527The cyclic permutation representation number of an integer partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000770The major index of an integer partition when read from bottom to top. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000260The radius of a connected graph. St001280The number of parts of an integer partition that are at least two. St001924The number of cells in an integer partition whose arm and leg length coincide. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001481The minimal height of a peak of a Dyck path. St000120The number of left tunnels of a Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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. St000100The number of linear extensions 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. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St001432The order dimension of the partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000456The monochromatic index of a connected graph. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001128The exponens consonantiae of a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001498The normalised height of a Nakayama algebra with magnitude 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001645The pebbling number of a connected graph. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000635The number of strictly order preserving maps of a poset into itself. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000284The Plancherel distribution on integer partitions. St001060The distinguishing index of a graph. St001568The smallest positive integer that does not appear twice in the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000344The number of strongly connected outdegree sequences of a graph. St000379The number of Hamiltonian cycles in a graph. St001368The number of vertices of maximal degree in a graph. 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. St000264The girth of a graph, which is not a tree. 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 $eA$. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001330The hat guessing number of a graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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