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Your data matches 169 different statistics following compositions of up to 3 maps.
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Matching statistic: St000307
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> 1
([],2)
=> 2
([(0,1)],2)
=> 1
([],3)
=> 4
([(1,2)],3)
=> 1
([(0,1),(0,2)],3)
=> 2
([(0,2),(2,1)],3)
=> 1
([(0,2),(1,2)],3)
=> 2
([],4)
=> 8
([(2,3)],4)
=> 2
([(1,2),(1,3)],4)
=> 3
([(0,1),(0,2),(0,3)],4)
=> 4
([(0,2),(0,3),(3,1)],4)
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,2),(2,3)],4)
=> 2
([(0,3),(3,1),(3,2)],4)
=> 2
([(1,3),(2,3)],4)
=> 3
([(0,3),(1,3),(3,2)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 4
([(0,3),(1,2)],4)
=> 3
([(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 3
([(0,3),(2,1),(3,2)],4)
=> 1
([(0,3),(1,2),(2,3)],4)
=> 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 1
([],2)
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> 1
([],3)
=> [2,2,2,2]
=> 4
([(1,2)],3)
=> [6]
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> 8
([(2,3)],4)
=> [6,6]
=> 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
Description
The length of the partition.
Matching statistic: St001280
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 1
([],2)
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> 1
([],3)
=> [2,2,2,2]
=> 4
([(1,2)],3)
=> [6]
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> 8
([(2,3)],4)
=> [6,6]
=> 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000147
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> 1
([],2)
=> [2,2]
=> [2,2]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 1
([],3)
=> [2,2,2,2]
=> [4,4]
=> 4
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> [8,8]
=> 8
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St000393
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00317: Integer partitions —odd parts⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 0 => 1
([],2)
=> [2,2]
=> 00 => 2
([(0,1)],2)
=> [3]
=> 1 => 1
([],3)
=> [2,2,2,2]
=> 0000 => 4
([(1,2)],3)
=> [6]
=> 0 => 1
([(0,1),(0,2)],3)
=> [3,2]
=> 10 => 2
([(0,2),(2,1)],3)
=> [4]
=> 0 => 1
([(0,2),(1,2)],3)
=> [3,2]
=> 10 => 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> 00000000 => 8
([(2,3)],4)
=> [6,6]
=> 00 => 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> 000 => 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1000 => 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 00 => 2
([(1,2),(2,3)],4)
=> [4,4]
=> 00 => 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 00 => 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> 000 => 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 00 => 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1000 => 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111 => 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 11 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 100 => 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1 => 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 => 1
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St001267
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00317: Integer partitions —odd parts⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 0 => 1
([],2)
=> [2,2]
=> 00 => 2
([(0,1)],2)
=> [3]
=> 1 => 1
([],3)
=> [2,2,2,2]
=> 0000 => 4
([(1,2)],3)
=> [6]
=> 0 => 1
([(0,1),(0,2)],3)
=> [3,2]
=> 10 => 2
([(0,2),(2,1)],3)
=> [4]
=> 0 => 1
([(0,2),(1,2)],3)
=> [3,2]
=> 10 => 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> 00000000 => 8
([(2,3)],4)
=> [6,6]
=> 00 => 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> 000 => 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1000 => 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 00 => 2
([(1,2),(2,3)],4)
=> [4,4]
=> 00 => 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 00 => 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> 000 => 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 00 => 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1000 => 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111 => 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 11 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 100 => 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1 => 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 => 1
Description
The length of the Lyndon factorization of the binary word.
The Lyndon factorization of a finite word w is its unique factorization as a non-increasing product of Lyndon words, i.e., $w = l_1\dots l_n$ where each $l_i$ is a Lyndon word and $l_1 \geq\dots\geq l_n$.
Matching statistic: St001437
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00317: Integer partitions —odd parts⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 0 => 1
([],2)
=> [2,2]
=> 00 => 2
([(0,1)],2)
=> [3]
=> 1 => 1
([],3)
=> [2,2,2,2]
=> 0000 => 4
([(1,2)],3)
=> [6]
=> 0 => 1
([(0,1),(0,2)],3)
=> [3,2]
=> 10 => 2
([(0,2),(2,1)],3)
=> [4]
=> 0 => 1
([(0,2),(1,2)],3)
=> [3,2]
=> 10 => 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> 00000000 => 8
([(2,3)],4)
=> [6,6]
=> 00 => 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> 000 => 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1000 => 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 00 => 2
([(1,2),(2,3)],4)
=> [4,4]
=> 00 => 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 00 => 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> 000 => 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 00 => 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1000 => 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111 => 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 11 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 100 => 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1 => 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 => 1
Description
The flex of a binary word.
This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
Matching statistic: St000519
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000519: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00317: Integer partitions —odd parts⟶ Binary words
St000519: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> 0 => 0 = 1 - 1
([],2)
=> [2,2]
=> 00 => 1 = 2 - 1
([(0,1)],2)
=> [3]
=> 1 => 0 = 1 - 1
([],3)
=> [2,2,2,2]
=> 0000 => 3 = 4 - 1
([(1,2)],3)
=> [6]
=> 0 => 0 = 1 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> 10 => 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> 0 => 0 = 1 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> 10 => 1 = 2 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> 00000000 => 7 = 8 - 1
([(2,3)],4)
=> [6,6]
=> 00 => 1 = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> 000 => 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1000 => 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> 1 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 00 => 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> 00 => 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 00 => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> 000 => 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 00 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1000 => 3 = 4 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111 => 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 11 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 100 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> 1 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 => 0 = 1 - 1
Description
The largest length of a factor maximising the subword complexity.
Let $p_w(n)$ be the number of distinct factors of length $n$. Then the statistic is the largest $n$ such that $p_w(n)$ is maximal:
$$
H_w = \max\{n: p_w(n)\text{ is maximal}\}
$$
A related statistic is the number of distinct factors of arbitrary length, also known as subword complexity, [[St000294]].
Matching statistic: St001918
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1]
=> 0 = 1 - 1
([],2)
=> [2,2]
=> [2,2]
=> 1 = 2 - 1
([(0,1)],2)
=> [3]
=> [1,1,1]
=> 0 = 1 - 1
([],3)
=> [2,2,2,2]
=> [4,4]
=> 3 = 4 - 1
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 1 = 2 - 1
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1]
=> 0 = 1 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 1 = 2 - 1
([],4)
=> [2,2,2,2,2,2,2,2]
=> [8,8]
=> 7 = 8 - 1
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> 1 = 2 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 2 = 3 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 3 = 4 - 1
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> 1 = 2 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 2 = 3 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1]
=> 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 1 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St000394
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
([],2)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
([(0,1)],2)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
([],3)
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 4
([(1,2)],3)
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([(0,2),(2,1)],3)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
([],4)
=> [2,2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> 8
([(2,3)],4)
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 2
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> 3
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 4
([(0,2),(0,3),(3,1)],4)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> 3
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 4
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
([(0,3),(2,1),(3,2)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
([(0,3),(1,2),(2,3)],4)
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
The following 159 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001814The number of partitions interlacing the given partition. St000439The position of the first down step of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000146The Andrews-Garvan crank of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000668The least common multiple of the parts of the partition. St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000548The number of different non-empty partial sums of an integer partition. St000507The number of ascents of a standard tableau. St000734The last entry in the first row of a standard tableau. St000143The largest repeated part of a partition. St000052The number of valleys of a Dyck path not on the x-axis. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000053The number of valleys of the Dyck path. St000025The number of initial rises of a Dyck path. St001746The coalition number of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000015The number of peaks of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000120The number of left tunnels of a Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001462The number of factors of a standard tableaux under concatenation. St001480The number of simple summands of the module J^2/J^3. St000260The radius of a connected graph. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001712The number of natural descents of a standard Young tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St001597The Frobenius rank of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001596The number of two-by-two squares inside a skew partition. St000453The number of distinct Laplacian eigenvalues of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001093The detour number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001341The number of edges in the center of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001691The number of kings in a graph. St001736The total number of cycles in a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001816Eigenvalues of the top-to-random operator acting on a simple module. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000633The size of the automorphism group of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St000455The second largest eigenvalue of a graph if it is integral. 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 $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000180The number of chains of a poset. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000299The number of nonisomorphic vertex-induced subtrees. 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. 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. St000081The number of edges of a graph. St000454The largest eigenvalue of a graph if it is integral. St001649The length of a longest trail in a graph. St001869The maximum cut size of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000418The number of Dyck paths that are weakly below a Dyck path. St000438The position of the last up step in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. 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. St001531Number of partial orders contained in the poset determined by the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001959The product of the heights of the peaks of a Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. 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. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St001360The number of covering relations in Young's lattice below a 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. St000145The Dyson rank of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001877Number of indecomposable injective modules with projective dimension 2. 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 $eA$. St001199The dominant dimension of $eAe$ for 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 $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001624The breadth of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001330The hat guessing number of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000264The girth of a graph, which is not a tree. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000100The number of linear extensions of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000656The number of cuts of a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001637The number of (upper) dissectors of a poset. 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. St000456The monochromatic index of a connected graph. St000741The Colin de Verdière graph invariant. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000284The Plancherel distribution on integer partitions. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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