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Your data matches 130 different statistics following compositions of up to 3 maps.
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Matching statistic: St000672
(load all 19 compositions to match this statistic)

Mapping

St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 0 = 1 - 1
[1,2] => 1 = 2 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 2 = 3 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 3 = 4 - 1
[1,2,4,3] => 2 = 3 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => 1 = 2 - 1
[2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => 2 = 3 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 2 = 3 - 1
[3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 1 = 2 - 1
[4,1,2,3] => 2 = 3 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 1 = 2 - 1
[4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => 1 = 2 - 1
[4,3,2,1] => 0 = 1 - 1

Description

The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is

$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$ for some

$(r,a,b)$ . This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

Matching statistic: St001773
(load all 8 compositions to match this statistic)

Mapping

Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001773: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1
[1,2] => [1,2] => 2
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 3
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => 2
[3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => 3
[1,3,4,2] => [1,3,4,2] => 3
[1,4,2,3] => [1,4,2,3] => 3
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 3
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 3
[2,3,4,1] => [2,3,4,1] => 3
[2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [3,1,2,4] => 3
[3,1,4,2] => [3,1,4,2] => 3
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [3,2,4,1] => 2
[3,4,1,2] => [3,4,1,2] => 3
[3,4,2,1] => [3,4,2,1] => 2
[4,1,2,3] => [4,1,2,3] => 3
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,2,3,1] => 2
[4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => 1

Description

The number of minimal elements in Bruhat order not less than the signed permutation. The minimal elements in question are biGrassmannian, that is both the element and its inverse have at most one descent. This is the size of the essential set of the signed permutation, see [1].

Matching statistic: St001489
(load all 14 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 0 = 1 - 1
[1,2] => [2,1] => 1 = 2 - 1
[2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [3,2,1] => 2 = 3 - 1
[1,3,2] => [3,1,2] => 1 = 2 - 1
[2,1,3] => [2,3,1] => 1 = 2 - 1
[2,3,1] => [2,1,3] => 1 = 2 - 1
[3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[1,2,4,3] => [4,3,1,2] => 2 = 3 - 1
[1,3,2,4] => [4,2,3,1] => 2 = 3 - 1
[1,3,4,2] => [4,2,1,3] => 2 = 3 - 1
[1,4,2,3] => [4,1,3,2] => 2 = 3 - 1
[1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[2,1,3,4] => [3,4,2,1] => 2 = 3 - 1
[2,1,4,3] => [3,4,1,2] => 1 = 2 - 1
[2,3,1,4] => [3,2,4,1] => 2 = 3 - 1
[2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[2,4,3,1] => [3,1,2,4] => 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[3,2,1,4] => [2,3,4,1] => 1 = 2 - 1
[3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => 0 = 1 - 1

Description

The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].

Matching statistic: St001615
(load all 2 compositions to match this statistic)

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001615: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,2,1] => ([],3)
 => ([],1)
 => 0 = 1 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,3,2,1] => ([],4)
 => ([],1)
 => 0 = 1 - 1

Description

The number of join prime elements of a lattice. An element

$x$ of a lattice

$L$ is join-prime (or coprime) if

$x \leq a \vee b$ implies

$x \leq a$ or

$x \leq b$ for every

$a, b \in L$ .

Matching statistic: St001617
(load all 2 compositions to match this statistic)

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001617: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,2,1] => ([],3)
 => ([],1)
 => 0 = 1 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,3,2,1] => ([],4)
 => ([],1)
 => 0 = 1 - 1

Description

The dimension of the space of valuations of a lattice. A valuation, or modular function, on a lattice

$L$ is a function

$v:L\mapsto\mathbb R$ satisfying

$v(a\vee b) + v(a\wedge b) = v(a) + v(b).$ It was shown by Birkhoff [1, thm. X.2], that a lattice with a positive valuation must be modular. This was sharpened by Fleischer and Traynor [2, thm. 1], which states that the modular functions on an arbitrary lattice are in bijection with the modular functions on its modular quotient [[Mp00196]]. Moreover, Birkhoff [1, thm. X.2] showed that the dimension of the space of modular functions equals the number of subsets of projective prime intervals.

Matching statistic: St001622
(load all 2 compositions to match this statistic)

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001622: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => 0 = 1 - 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1] => ([],2)
 => ([],1)
 => 0 = 1 - 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,2,1] => ([],3)
 => ([],1)
 => 0 = 1 - 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 2 - 1
[4,3,2,1] => ([],4)
 => ([],1)
 => 0 = 1 - 1

Description

The number of join-irreducible elements of a lattice. An element

$j$ of a lattice

$L$ is '''join irreducible''' if it is not the least element and if

$j=x\vee y$ , then

$j\in\{x,y\}$ for all

$x,y\in L$ .

Matching statistic: St001812
(load all 5 compositions to match this statistic)

Mapping

Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => ([],1)
 => 0 = 1 - 1
[1,2] => [2,1] => ([(0,1)],2)
 => 1 = 2 - 1
[2,1] => [1,2] => ([],2)
 => 0 = 1 - 1
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[2,3,1] => [1,3,2] => ([(1,2)],3)
 => 1 = 2 - 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
 => 1 = 2 - 1
[3,2,1] => [1,2,3] => ([],3)
 => 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1 = 2 - 1
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2 = 3 - 1
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
 => 1 = 2 - 1
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
 => 1 = 2 - 1
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
 => 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => ([],4)
 => 0 = 1 - 1

Description

The biclique partition number of a graph. The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph

$K_n$ has biclique partition number

$n - 1$ .

Matching statistic: St000528

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000528: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => 1

Description

The height of a poset. This equals the rank of the poset [[St000080]] plus one.

Matching statistic: St000912

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => 1

Description

The number of maximal antichains in a poset.

Matching statistic: St001343

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001343: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => ([],1)
 => ([],1)
 => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1] => ([],2)
 => ([],1)
 => ([],1)
 => 1
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,1] => ([],3)
 => ([],1)
 => ([],1)
 => 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[4,3,2,1] => ([],4)
 => ([],1)
 => ([],1)
 => 1

Description

The dimension of the reduced incidence algebra of a poset. The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets. Thus, this statistic returns the number of non-isomorphic intervals of the poset.

The following 120 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St000362The size of a minimal vertex cover of a graph. St001613The binary logarithm of the size of the center of a lattice. St001782The order of rowmotion on the set of order ideals of a poset. St001621The number of atoms of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. 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. St001626The number of maximal proper sublattices of a lattice. St000454The largest eigenvalue of a graph if it is integral. St000932The number of occurrences of the pattern UDU in a Dyck path. St000567The sum of the products of all pairs of parts. 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. St001875The number of simple modules with projective dimension at most 1. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001877Number of indecomposable injective modules with projective dimension 2. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001961The sum of the greatest common divisors of all pairs of parts. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000761The number of ascents in an integer composition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000741The Colin de Verdière graph invariant. St000456The monochromatic index of a connected graph. St001330The hat guessing number of a graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000997The even-odd crank of an integer partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. 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. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The 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 vertex labelled 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000264The girth of a graph, which is not a tree. St000736The last entry in the first row of a semistandard tableau. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation.