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Your data matches 283 different statistics following compositions of up to 3 maps.
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Matching statistic: St001399
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> 2
([(0,2),(2,1)],3)
=> 1
([(0,2),(1,2)],3)
=> 2
([(0,2),(0,3),(3,1)],4)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,3),(2,1),(3,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
Description
The distinguishing number of a poset.
This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring.
See also [[St000469]], which is the same concept for graphs.
Matching statistic: St000183
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(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 1
([],2)
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> 1
([(1,2)],3)
=> [6]
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000755
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000755: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 1
([],2)
=> [2]
=> 2
([(0,1)],2)
=> [1]
=> 1
([(1,2)],3)
=> [3]
=> 1
([(0,1),(0,2)],3)
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [1]
=> 1
([(0,2),(1,2)],3)
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [3]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 1
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence $$f(n)=\sum_{p\in\lambda} f(n-p).$$ This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition $(2,1)$ corresponds to the recurrence $f(n)=f(n-1)+f(n-2)$ with associated characteristic polynomial $x^2-x-1$, which has two real roots.
Matching statistic: St000481
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000481: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 0 = 1 - 1
([],2)
=> [2,2]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> 0 = 1 - 1
([(1,2)],3)
=> [6]
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> 0 = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> 0 = 1 - 1
Description
The number of upper covers of a partition in dominance order.
Matching statistic: St001092
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 0 = 1 - 1
([],2)
=> [2]
=> 1 = 2 - 1
([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(1,2)],3)
=> [3]
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> [2]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [2]
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [2]
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0 = 1 - 1
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St001587
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [1]
=> 0 = 1 - 1
([],2)
=> [2]
=> 1 = 2 - 1
([(0,1)],2)
=> [1]
=> 0 = 1 - 1
([(1,2)],3)
=> [3]
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> [2]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [1]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [2]
=> 1 = 2 - 1
([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [2]
=> 1 = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 1 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1 = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 1 = 2 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0 = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 1 = 2 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0 = 1 - 1
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
Matching statistic: St000318
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(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> []
=> 1
([],2)
=> [2,2]
=> [2]
=> 2
([(0,1)],2)
=> [3]
=> []
=> 1
([(1,2)],3)
=> [6]
=> []
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> []
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> []
=> 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [2]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> []
=> 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000531
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> []
=> 1
([],2)
=> [2,2]
=> [2]
=> 2
([(0,1)],2)
=> [3]
=> []
=> 1
([(1,2)],3)
=> [6]
=> []
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> []
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> 2
([(0,2),(0,3),(3,1)],4)
=> [7]
=> []
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> 2
([(0,3),(2,1),(3,2)],4)
=> [5]
=> []
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> []
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> [6]
=> []
=> 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [2]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [7]
=> []
=> 1
Description
The leading coefficient of the rook polynomial of an integer partition.
Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.
Matching statistic: St000723
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],1)
=> ([],1)
=> 1
([],2)
=> ([(0,1)],2)
=> ([],2)
=> 2
([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(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)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],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
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(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)
=> 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)
=> 2
([(0,3),(3,4),(4,1),(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)
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The maximal cardinality of a set of vertices with the same neighbourhood in a graph.
The set of so called mating graphs, for which this statistic equals $1$, is enumerated by [1].
Matching statistic: St000759
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> ([],1)
=> []
=> 1
([],2)
=> ([(0,1)],2)
=> [1]
=> 2
([(0,1)],2)
=> ([],2)
=> []
=> 1
([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> [2]
=> 1
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> [1]
=> 2
([(0,2),(2,1)],3)
=> ([],3)
=> []
=> 1
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [1]
=> 2
([(0,2),(0,3),(3,1)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> [1]
=> 2
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 2
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [1]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> [1,1]
=> 2
([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> []
=> 1
([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [2]
=> 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> [1]
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> [1]
=> 2
([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> []
=> 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> [1]
=> 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> []
=> 1
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
The following 273 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000920The logarithmic height of a Dyck path. St001597The Frobenius rank of a skew partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001955The number of natural descents for set-valued two row standard Young tableaux. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000159The number of distinct parts of the integer partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000340The number of non-final maximal constant sub-paths of length greater than one. St000480The number of lower covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000547The number of even non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000897The number of different multiplicities of parts of an integer partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001826The maximal number of leaves on a vertex of a graph. St000024The number of double up and double down steps of a Dyck path. St000291The number of descents of a binary word. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000617The number of global maxima of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000874The position of the last double rise in a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001471The magnitude of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001733The number of weak left to right maxima of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000150The floored half-sum of the multiplicities of a partition. St000271The chromatic index of a graph. St000306The bounce count of a Dyck path. St000377The dinv defect of an integer partition. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001484The number of singletons of an integer partition. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001809The index of the step at the first peak of maximal height in a Dyck path. St001956The comajor index for set-valued two-row standard Young tableaux. St000439The position of the first down step of a Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000307The number of rowmotion orbits of a poset. St000640The rank of the largest boolean interval in a poset. St001568The smallest positive integer that does not appear twice in the partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000454The largest eigenvalue of a graph if it is integral. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001273The projective dimension of the first term in an injective coresolution of the regular module. St000015The number of peaks of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001644The dimension of a graph. St000331The number of upper interactions of a Dyck path. St000443The number of long tunnels of a Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 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. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000455The second largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000268The number of strongly connected orientations of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St001073The number of nowhere zero 3-flows of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St000351The determinant of the adjacency matrix of a graph. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000096The number of spanning trees of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000450The number of edges minus the number of vertices plus 2 of a graph. St000658The number of rises of length 2 of a Dyck path. St000948The chromatic discriminant of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000302The determinant of the distance matrix of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000474Dyson's crank of a partition. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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 $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001933The largest multiplicity of a part in an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001564The value of the forgotten symmetric functions when all variables set to 1. St001563The value of the power-sum symmetric function evaluated at 1. St000928The sum of the coefficients of the character polynomial of an integer partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000741The Colin de Verdière graph invariant. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001330The hat guessing number of a graph. 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 $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. 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. St000264The girth of a graph, which is not a tree. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000456The monochromatic index of a connected graph. 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. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St000939The number of characters of the symmetric group whose value on the partition is positive. 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. St000567The sum of the products of all pairs of parts. 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. St001060The distinguishing index of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. 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. St001118The acyclic chromatic index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001570The minimal number of edges to add to make a graph Hamiltonian. St001545The second Elser number of a connected graph. St000464The Schultz index of a connected graph. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000934The 2-degree of an integer partition. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000699The toughness times the least common multiple of 1,.
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