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Your data matches 270 different statistics following compositions of up to 3 maps.
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Matching statistic: St001397
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Mp00013: Binary trees —to poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
Number of pairs of incomparable elements in a finite poset.
For a finite poset $(P,\leq)$, this is the number of unordered pairs $\{x,y\} \in \binom{P}{2}$ with $x \not\leq y$ and $y \not\leq x$.
Matching statistic: St001268
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
St001268: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001268: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The size of the largest ordinal summand in the poset.
The ordinal sum of two posets $P$ and $Q$ is the poset having elements $(p,0)$ and $(q,1)$ for $p\in P$ and $q\in Q$, and relations $(a,0) < (b,0)$ if $a < b$ in $P$, $(a,1) < (b,1)$ if $a < b$ in $Q$, and $(a,0) < (b,1)$.
This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.
Matching statistic: St001779
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
St001779: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001779: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 1 = 0 + 1
[.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The order of promotion on the set of linear extensions of a poset.
Matching statistic: St000081
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The number of edges of a graph.
Matching statistic: St000145
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> [1]
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> [1]
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
Description
The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$
Matching statistic: St000171
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> [1]
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> [1]
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> [1]
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> [1]
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [1]
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 0
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000987
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The number of positive eigenvalues of the Laplacian matrix of the graph.
This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St001117
Values
[.,.]
=> ([],1)
=> ([],1)
=> 0
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 0
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 0
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 1
[[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
Description
The game chromatic index of a graph.
Two players, Alice and Bob, take turns colouring properly any uncolored edge of the graph. Alice begins. If it is not possible for either player to colour a edge, then Bob wins. If the graph is completely colored, Alice wins.
The game chromatic index is the smallest number of colours such that Alice has a winning strategy.
The following 260 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001120The length of a longest path in a graph. St001341The number of edges in the center of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. 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. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001649The length of a longest trail in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000086The number of subgraphs. St000088The row sums of the character table of the symmetric group. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000228The size of a partition. St000299The number of nonisomorphic vertex-induced subtrees. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000384The maximal part of the shifted composition of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000468The Hosoya index of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000667The greatest common divisor of the parts of the partition. St000784The maximum of the length and the largest part of the integer partition. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000935The number of ordered refinements of an integer partition. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001360The number of covering relations in Young's lattice below a partition. St001367The smallest number which does not occur as degree of a vertex in a graph. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number 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. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001674The number of vertices of the largest induced star graph in the graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001725The harmonious chromatic number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St001400The total number of Littlewood-Richardson tableaux of given shape. St001814The number of partitions interlacing the given partition. St000220The number of occurrences of the pattern 132 in a permutation. St000225Difference between largest and smallest parts in a partition. St000377The dinv defect of an integer partition. St000448The number of pairs of vertices of a graph with distance 2. St001091The number of parts in an integer partition whose next smaller part has the same size. St001176The size of a partition minus its first part. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000010The length of the partition. St000071The number of maximal chains in a poset. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000255The number of reduced Kogan faces with the permutation as type. St000346The number of coarsenings of a partition. St000456The monochromatic index of a connected graph. St000475The number of parts equal to 1 in a partition. St000548The number of different non-empty partial sums of an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000820The number of compositions obtained by rotating the composition. St000867The sum of the hook lengths in the first row of an integer partition. St000909The number of maximal chains of maximal size in a poset. St001127The sum of the squares of the parts of a partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001917The order of toric promotion on the set of labellings of a graph. St001933The largest multiplicity of a part in an integer partition. St000849The number of 1/3-balanced pairs in a poset. St000100The number of linear extensions of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000940The number of characters of the symmetric group whose value on the partition is zero. St001388The number of non-attacking neighbors of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000045The number of linear extensions of a binary tree. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000422The energy of a graph, if it is integral. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000095The number of triangles of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St000450The number of edges minus the number of vertices plus 2 of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000850The number of 1/2-balanced pairs in a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001399The distinguishing number of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St000464The Schultz index of a connected graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001172The number of 1-rises at odd height of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St001118The acyclic chromatic index of a graph. St000477The weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000928The sum of the coefficients of the character polynomial of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St000409The number of pitchforks in a binary tree. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000214The number of adjacencies of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001481The minimal height of a peak of a Dyck path. St001949The rigidity index of a graph. St000906The length of the shortest maximal chain in a poset. St000091The descent variation of a composition. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000323The minimal crossing number of a graph. St000365The number of double ascents of a permutation. St000370The genus of a graph. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000709The number of occurrences of 14-2-3 or 14-3-2. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001549The number of restricted non-inversions between exceedances. St001857The number of edges in the reduced word graph of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000025The number of initial rises of a Dyck path. St000352The Elizalde-Pak rank of a permutation. St000544The cop number of a graph. St000617The number of global maxima of a Dyck path. St000776The maximal multiplicity of an eigenvalue in a graph. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001272The number of graphs with the same degree sequence. St001340The cardinality of a minimal non-edge isolating set of a graph. St001405The number of bonds in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001510The number of self-evacuating linear extensions of a finite poset. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001694The number of maximal dissociation sets in a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000258The burning number of a graph. St000273The domination number of a graph. St000335The difference of lower and upper interactions. St000842The breadth of a permutation. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. 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 $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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 $eA$. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001530The depth of a Dyck path. St001829The common independence number of a graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St001271The competition number of a graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000893The number of distinct diagonal sums of an alternating sign matrix. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers.
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