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Your data matches 332 different statistics following compositions of up to 3 maps.
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Matching statistic: St001958
(load all 831 compositions to match this statistic)
(load all 831 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1 = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1 = 0 + 1
Description
The degree of the polynomial interpolating the values of a permutation.
Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$.
This statistic records the degree of $p$.
Matching statistic: St000222
(load all 101 compositions to match this statistic)
(load all 101 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000222: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => 0
Description
The number of alignments in the permutation.
Matching statistic: St000425
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000425: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000425: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 0
Description
The number of occurrences of the pattern 132 or of the pattern 213 in a permutation.
Matching statistic: St000457
(load all 61 compositions to match this statistic)
(load all 61 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000457: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000457: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => 0
[1,0,1,1,0,0]
=> [2,1,3] => [3,2,1] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,4,1,2] => 0
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,4,2,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,1,3] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [2,4,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,3,4,2] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [3,4,5,6,1,2] => 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [2,3,4,5,6,1] => 0
Description
The number of occurrences of one of the patterns 132, 213 or 321 in a permutation.
According to [1], this statistic was studied by Doron Gepner in the context of conformal field theory.
Matching statistic: St000848
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000848: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00209: Permutations —pattern poset⟶ Posets
St000848: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 0
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The balance constant multiplied with the number of linear extensions of a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion $P(x,y)$ of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. The balance constant of a poset is $\max\min(P(x,y), P(y,x)).$
Kislitsyn [1] conjectured that every poset which is not a chain is $1/3$-balanced. Brightwell, Felsner and Trotter [2] show that it is at least $(1-\sqrt 5)/10$-balanced.
Olson and Sagan [3] exhibit various posets that are $1/2$-balanced.
Matching statistic: St000849
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000849: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00209: Permutations —pattern poset⟶ Posets
St000849: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 0
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The number of 1/3-balanced pairs in a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$.
Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains.
Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a $1/3$-balanced pair.
Matching statistic: St001633
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 0
[1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 0
[1,0,1,1,0,0]
=> [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000003
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [2,1] => [2]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6]
=> 1 = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 1 = 0 + 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St000277
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1 = 0 + 1
[1,1,0,0]
=> [1,1,0,0]
=> [2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 1 = 0 + 1
Description
The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' [[Mp00071]] from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals
$$
\sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell},
$$
where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]
Matching statistic: St000530
(load all 70 compositions to match this statistic)
(load all 70 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000530: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000530: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => 1 = 0 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 1 = 0 + 1
Description
The number of permutations with the same descent word as the given permutation.
The descent word of a permutation is the binary word given by [[Mp00109]]. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map [[Mp00109]] containing a given permutation.
This statistic appears as ''up-down analysis'' in statistical applications in genetics, see [1] and the references therein.
The following 322 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000543The size of the conjugacy class of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000057The Shynar inversion number of a standard tableau. St000220The number of occurrences of the pattern 132 in a permutation. St000293The number of inversions of a binary word. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000538The number of even inversions of a permutation. St000682The Grundy value of Welter's game on a binary word. St000836The number of descents of distance 2 of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001306The number of induced paths on four vertices in a graph. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001727The number of invisible inversions of a permutation. St001910The height of the middle non-run of a Dyck path. St000047The number of standard immaculate tableaux of a given shape. St000075The orbit size of a standard tableau under promotion. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000529The number of permutations whose descent word is the given binary word. St000626The minimal period of a binary word. St001090The number of pop-stack-sorts needed to sort a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001313The number of Dyck paths above the lattice path given by a binary word. St001595The number of standard Young tableaux of the skew partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001120The length of a longest path in a graph. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001535The number of cyclic alignments of a permutation. St000789The number of crossing-similar perfect matchings of a perfect matching. 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. St001510The number of self-evacuating linear extensions of a finite poset. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000045The number of linear extensions of a binary tree. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001822The number of alignments of a signed permutation. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000035The number of left outer peaks of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000628The balance of a binary word. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St000251The number of nonsingleton blocks of a set partition. St001115The number of even descents of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000717The number of ordinal summands of a poset. St000871The number of very big ascents of a permutation. St001964The interval resolution global dimension of a poset. St000834The number of right outer peaks of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001623The number of doubly irreducible elements of a lattice. 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. St001893The flag descent of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000233The number of nestings of a set partition. St000259The diameter of a connected graph. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000516The number of stretching pairs of a permutation. St000648The number of 2-excedences of a permutation. St000650The number of 3-rises of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000872The number of very big descents of a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001403The number of vertical separators in a permutation. St001715The number of non-records in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000023The number of inner peaks of a permutation. St000254The nesting number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000388The number of orbits of vertices of a graph under automorphisms. St000570The Edelman-Greene number of a permutation. St000779The tier of a permutation. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St001114The number of odd descents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001469The holeyness of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001729The number of visible descents of a permutation. St001768The number of reduced words of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001935The number of ascents in a parking function. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000099The number of valleys of a permutation, including the boundary. St000308The height of the tree associated to a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000630The length of the shortest palindromic decomposition of a binary word. St001488The number of corners of a skew partition. St001517The length of a longest pair of twins in a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001667The maximal size of a pair of weak twins for a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001769The reflection length of a signed permutation. St000503The maximal difference between two elements in a common block. 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 $eA$. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000735The last entry on the main diagonal of a standard tableau. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001645The pebbling number of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001060The distinguishing index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St000632The jump number of the poset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St000527The width of the poset. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000845The maximal number of elements covered by an element in a poset. 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. St001118The acyclic chromatic index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000181The number of connected components of the Hasse diagram for the poset. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001890The maximum magnitude of the Möbius function of a poset. St000058The order of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000893The number of distinct diagonal sums of an alternating sign matrix. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000264The girth of a graph, which is not a tree. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000422The energy of a graph, if it is integral. St000124The cardinality of the preimage of the Simion-Schmidt map. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000478Another weight of a partition according to Alladi. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000509The diagonal index (content) of a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000929The constant term of the character polynomial of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001615The number of join prime elements of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001846The number of elements which do not have a complement in the lattice. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000667The greatest common divisor of the parts of the partition. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000909The number of maximal chains of maximal size in a poset. 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. St000944The 3-degree of an integer partition. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001587Half of the largest even part of an integer partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001613The binary logarithm of the size of the center of a lattice. St001657The number of twos in an integer partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000137The Grundy value of an integer partition. St000306The bounce count of a Dyck path. St000474Dyson's crank of a partition. St000477The 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. St000668The least common multiple of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000770The major index of an integer partition when read from bottom to top. St000934The 2-degree of an integer partition. St000997The even-odd crank of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001525The number of symmetric hooks on the diagonal of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001616The number of neutral elements in a lattice. St001644The dimension of a graph. St001720The minimal length of a chain of small intervals in a lattice. 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. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001389The number of partitions of the same length below the given integer partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000567The sum of the products of all pairs of parts. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001527The cyclic permutation representation number of an integer partition. St001541The Gini index of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. 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. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001378The product of the cohook lengths of the integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St001627The number of coloured connected 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. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001943The sum of the squares of the hook lengths of an integer partition. 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. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. 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. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001763The Hurwitz number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition.
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