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Your data matches 369 different statistics following compositions of up to 3 maps.
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Matching statistic: St000662
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0 = 1 - 1
[-] => [1] => 0 = 1 - 1
[+,+] => [1,2] => 0 = 1 - 1
[-,+] => [1,2] => 0 = 1 - 1
[+,-] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 1 = 2 - 1
[+,+,+] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [1,2,3] => 0 = 1 - 1
[+,-,+] => [1,2,3] => 0 = 1 - 1
[+,+,-] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [1,2,3] => 0 = 1 - 1
[-,+,-] => [1,2,3] => 0 = 1 - 1
[+,-,-] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,3,2] => 1 = 2 - 1
[-,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,+] => [2,1,3] => 1 = 2 - 1
[2,1,-] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [2,3,1] => 1 = 2 - 1
[3,1,2] => [3,1,2] => 1 = 2 - 1
[+,+,+,+] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [1,2,3,4] => 0 = 1 - 1
[+,-,+,+] => [1,2,3,4] => 0 = 1 - 1
[+,+,-,+] => [1,2,3,4] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [1,2,3,4] => 0 = 1 - 1
[-,+,-,+] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,-] => [1,2,3,4] => 0 = 1 - 1
[+,-,-,+] => [1,2,3,4] => 0 = 1 - 1
[+,-,+,-] => [1,2,3,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,-] => [1,2,3,4] => 0 = 1 - 1
[-,+,-,-] => [1,2,3,4] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => 1 = 2 - 1
[-,+,4,3] => [1,2,4,3] => 1 = 2 - 1
[+,-,4,3] => [1,2,4,3] => 1 = 2 - 1
[-,-,4,3] => [1,2,4,3] => 1 = 2 - 1
[+,3,2,+] => [1,3,2,4] => 1 = 2 - 1
[-,3,2,+] => [1,3,2,4] => 1 = 2 - 1
[+,3,2,-] => [1,3,2,4] => 1 = 2 - 1
[-,3,2,-] => [1,3,2,4] => 1 = 2 - 1
[+,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[-,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[+,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[-,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[2,1,+,+] => [2,1,3,4] => 1 = 2 - 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St001761
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => 0 = 1 - 1
[-] => [1] => 0 = 1 - 1
[+,+] => [1,2] => 0 = 1 - 1
[-,+] => [1,2] => 0 = 1 - 1
[+,-] => [1,2] => 0 = 1 - 1
[-,-] => [1,2] => 0 = 1 - 1
[2,1] => [2,1] => 1 = 2 - 1
[+,+,+] => [1,2,3] => 0 = 1 - 1
[-,+,+] => [1,2,3] => 0 = 1 - 1
[+,-,+] => [1,2,3] => 0 = 1 - 1
[+,+,-] => [1,2,3] => 0 = 1 - 1
[-,-,+] => [1,2,3] => 0 = 1 - 1
[-,+,-] => [1,2,3] => 0 = 1 - 1
[+,-,-] => [1,2,3] => 0 = 1 - 1
[-,-,-] => [1,2,3] => 0 = 1 - 1
[+,3,2] => [1,3,2] => 1 = 2 - 1
[-,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,+] => [2,1,3] => 1 = 2 - 1
[2,1,-] => [2,1,3] => 1 = 2 - 1
[2,3,1] => [2,3,1] => 1 = 2 - 1
[3,1,2] => [3,1,2] => 1 = 2 - 1
[+,+,+,+] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,+] => [1,2,3,4] => 0 = 1 - 1
[+,-,+,+] => [1,2,3,4] => 0 = 1 - 1
[+,+,-,+] => [1,2,3,4] => 0 = 1 - 1
[+,+,+,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,+] => [1,2,3,4] => 0 = 1 - 1
[-,+,-,+] => [1,2,3,4] => 0 = 1 - 1
[-,+,+,-] => [1,2,3,4] => 0 = 1 - 1
[+,-,-,+] => [1,2,3,4] => 0 = 1 - 1
[+,-,+,-] => [1,2,3,4] => 0 = 1 - 1
[+,+,-,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,+] => [1,2,3,4] => 0 = 1 - 1
[-,-,+,-] => [1,2,3,4] => 0 = 1 - 1
[-,+,-,-] => [1,2,3,4] => 0 = 1 - 1
[+,-,-,-] => [1,2,3,4] => 0 = 1 - 1
[-,-,-,-] => [1,2,3,4] => 0 = 1 - 1
[+,+,4,3] => [1,2,4,3] => 1 = 2 - 1
[-,+,4,3] => [1,2,4,3] => 1 = 2 - 1
[+,-,4,3] => [1,2,4,3] => 1 = 2 - 1
[-,-,4,3] => [1,2,4,3] => 1 = 2 - 1
[+,3,2,+] => [1,3,2,4] => 1 = 2 - 1
[-,3,2,+] => [1,3,2,4] => 1 = 2 - 1
[+,3,2,-] => [1,3,2,4] => 1 = 2 - 1
[-,3,2,-] => [1,3,2,4] => 1 = 2 - 1
[+,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[-,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[+,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[-,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[2,1,+,+] => [2,1,3,4] => 1 = 2 - 1
Description
The maximal multiplicity of a letter in a reduced word of a permutation.
For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Matching statistic: St000010
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1]
=> 1
[-] => [1] => [1]
=> 1
[+,+] => [1,2] => [2]
=> 1
[-,+] => [1,2] => [2]
=> 1
[+,-] => [1,2] => [2]
=> 1
[-,-] => [1,2] => [2]
=> 1
[2,1] => [2,1] => [1,1]
=> 2
[+,+,+] => [1,2,3] => [3]
=> 1
[-,+,+] => [1,2,3] => [3]
=> 1
[+,-,+] => [1,2,3] => [3]
=> 1
[+,+,-] => [1,2,3] => [3]
=> 1
[-,-,+] => [1,2,3] => [3]
=> 1
[-,+,-] => [1,2,3] => [3]
=> 1
[+,-,-] => [1,2,3] => [3]
=> 1
[-,-,-] => [1,2,3] => [3]
=> 1
[+,3,2] => [1,3,2] => [2,1]
=> 2
[-,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,+] => [2,1,3] => [2,1]
=> 2
[2,1,-] => [2,1,3] => [2,1]
=> 2
[2,3,1] => [2,3,1] => [2,1]
=> 2
[3,1,2] => [3,1,2] => [2,1]
=> 2
[+,+,+,+] => [1,2,3,4] => [4]
=> 1
[-,+,+,+] => [1,2,3,4] => [4]
=> 1
[+,-,+,+] => [1,2,3,4] => [4]
=> 1
[+,+,-,+] => [1,2,3,4] => [4]
=> 1
[+,+,+,-] => [1,2,3,4] => [4]
=> 1
[-,-,+,+] => [1,2,3,4] => [4]
=> 1
[-,+,-,+] => [1,2,3,4] => [4]
=> 1
[-,+,+,-] => [1,2,3,4] => [4]
=> 1
[+,-,-,+] => [1,2,3,4] => [4]
=> 1
[+,-,+,-] => [1,2,3,4] => [4]
=> 1
[+,+,-,-] => [1,2,3,4] => [4]
=> 1
[-,-,-,+] => [1,2,3,4] => [4]
=> 1
[-,-,+,-] => [1,2,3,4] => [4]
=> 1
[-,+,-,-] => [1,2,3,4] => [4]
=> 1
[+,-,-,-] => [1,2,3,4] => [4]
=> 1
[-,-,-,-] => [1,2,3,4] => [4]
=> 1
[+,+,4,3] => [1,2,4,3] => [3,1]
=> 2
[-,+,4,3] => [1,2,4,3] => [3,1]
=> 2
[+,-,4,3] => [1,2,4,3] => [3,1]
=> 2
[-,-,4,3] => [1,2,4,3] => [3,1]
=> 2
[+,3,2,+] => [1,3,2,4] => [3,1]
=> 2
[-,3,2,+] => [1,3,2,4] => [3,1]
=> 2
[+,3,2,-] => [1,3,2,4] => [3,1]
=> 2
[-,3,2,-] => [1,3,2,4] => [3,1]
=> 2
[+,3,4,2] => [1,3,4,2] => [3,1]
=> 2
[-,3,4,2] => [1,3,4,2] => [3,1]
=> 2
[+,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[-,4,2,3] => [1,4,2,3] => [3,1]
=> 2
[2,1,+,+] => [2,1,3,4] => [3,1]
=> 2
Description
The length of the partition.
Matching statistic: St000058
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 1
[-] => [1] => [1] => 1
[+,+] => [1,2] => [1,2] => 1
[-,+] => [1,2] => [1,2] => 1
[+,-] => [1,2] => [1,2] => 1
[-,-] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 2
[+,+,+] => [1,2,3] => [1,2,3] => 1
[-,+,+] => [1,2,3] => [1,2,3] => 1
[+,-,+] => [1,2,3] => [1,2,3] => 1
[+,+,-] => [1,2,3] => [1,2,3] => 1
[-,-,+] => [1,2,3] => [1,2,3] => 1
[-,+,-] => [1,2,3] => [1,2,3] => 1
[+,-,-] => [1,2,3] => [1,2,3] => 1
[-,-,-] => [1,2,3] => [1,2,3] => 1
[+,3,2] => [1,3,2] => [1,3,2] => 2
[-,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,+] => [2,1,3] => [2,1,3] => 2
[2,1,-] => [2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => [3,2,1] => 2
[3,1,2] => [3,1,2] => [3,2,1] => 2
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => 1
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => 1
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => 1
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => 1
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => 1
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => 1
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => 1
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => 1
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => 1
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => 1
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => 1
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => 2
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => 2
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => 2
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => 2
[+,3,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[-,3,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[+,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2
[-,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => 2
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000062
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1] => 1
[-] => [1] => [1] => 1
[+,+] => [1,2] => [2,1] => 1
[-,+] => [1,2] => [2,1] => 1
[+,-] => [1,2] => [2,1] => 1
[-,-] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 2
[+,+,+] => [1,2,3] => [3,2,1] => 1
[-,+,+] => [1,2,3] => [3,2,1] => 1
[+,-,+] => [1,2,3] => [3,2,1] => 1
[+,+,-] => [1,2,3] => [3,2,1] => 1
[-,-,+] => [1,2,3] => [3,2,1] => 1
[-,+,-] => [1,2,3] => [3,2,1] => 1
[+,-,-] => [1,2,3] => [3,2,1] => 1
[-,-,-] => [1,2,3] => [3,2,1] => 1
[+,3,2] => [1,3,2] => [2,3,1] => 2
[-,3,2] => [1,3,2] => [2,3,1] => 2
[2,1,+] => [2,1,3] => [3,1,2] => 2
[2,1,-] => [2,1,3] => [3,1,2] => 2
[2,3,1] => [2,3,1] => [2,1,3] => 2
[3,1,2] => [3,1,2] => [1,3,2] => 2
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => 1
[-,+,+,+] => [1,2,3,4] => [4,3,2,1] => 1
[+,-,+,+] => [1,2,3,4] => [4,3,2,1] => 1
[+,+,-,+] => [1,2,3,4] => [4,3,2,1] => 1
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => 1
[-,-,+,+] => [1,2,3,4] => [4,3,2,1] => 1
[-,+,-,+] => [1,2,3,4] => [4,3,2,1] => 1
[-,+,+,-] => [1,2,3,4] => [4,3,2,1] => 1
[+,-,-,+] => [1,2,3,4] => [4,3,2,1] => 1
[+,-,+,-] => [1,2,3,4] => [4,3,2,1] => 1
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => 1
[-,-,-,+] => [1,2,3,4] => [4,3,2,1] => 1
[-,-,+,-] => [1,2,3,4] => [4,3,2,1] => 1
[-,+,-,-] => [1,2,3,4] => [4,3,2,1] => 1
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => 1
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => 1
[+,+,4,3] => [1,2,4,3] => [3,4,2,1] => 2
[-,+,4,3] => [1,2,4,3] => [3,4,2,1] => 2
[+,-,4,3] => [1,2,4,3] => [3,4,2,1] => 2
[-,-,4,3] => [1,2,4,3] => [3,4,2,1] => 2
[+,3,2,+] => [1,3,2,4] => [4,2,3,1] => 2
[-,3,2,+] => [1,3,2,4] => [4,2,3,1] => 2
[+,3,2,-] => [1,3,2,4] => [4,2,3,1] => 2
[-,3,2,-] => [1,3,2,4] => [4,2,3,1] => 2
[+,3,4,2] => [1,3,4,2] => [3,2,4,1] => 2
[-,3,4,2] => [1,3,4,2] => [3,2,4,1] => 2
[+,4,2,3] => [1,4,2,3] => [2,4,3,1] => 2
[-,4,2,3] => [1,4,2,3] => [2,4,3,1] => 2
[2,1,+,+] => [2,1,3,4] => [4,3,1,2] => 2
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000097
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
=> 1
[-] => [1] => ([],1)
=> 1
[+,+] => [1,2] => ([],2)
=> 1
[-,+] => [1,2] => ([],2)
=> 1
[+,-] => [1,2] => ([],2)
=> 1
[-,-] => [1,2] => ([],2)
=> 1
[2,1] => [2,1] => ([(0,1)],2)
=> 2
[+,+,+] => [1,2,3] => ([],3)
=> 1
[-,+,+] => [1,2,3] => ([],3)
=> 1
[+,-,+] => [1,2,3] => ([],3)
=> 1
[+,+,-] => [1,2,3] => ([],3)
=> 1
[-,-,+] => [1,2,3] => ([],3)
=> 1
[-,+,-] => [1,2,3] => ([],3)
=> 1
[+,-,-] => [1,2,3] => ([],3)
=> 1
[-,-,-] => [1,2,3] => ([],3)
=> 1
[+,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[-,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[2,1,+] => [2,1,3] => ([(1,2)],3)
=> 2
[2,1,-] => [2,1,3] => ([(1,2)],3)
=> 2
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[+,+,+,+] => [1,2,3,4] => ([],4)
=> 1
[-,+,+,+] => [1,2,3,4] => ([],4)
=> 1
[+,-,+,+] => [1,2,3,4] => ([],4)
=> 1
[+,+,-,+] => [1,2,3,4] => ([],4)
=> 1
[+,+,+,-] => [1,2,3,4] => ([],4)
=> 1
[-,-,+,+] => [1,2,3,4] => ([],4)
=> 1
[-,+,-,+] => [1,2,3,4] => ([],4)
=> 1
[-,+,+,-] => [1,2,3,4] => ([],4)
=> 1
[+,-,-,+] => [1,2,3,4] => ([],4)
=> 1
[+,-,+,-] => [1,2,3,4] => ([],4)
=> 1
[+,+,-,-] => [1,2,3,4] => ([],4)
=> 1
[-,-,-,+] => [1,2,3,4] => ([],4)
=> 1
[-,-,+,-] => [1,2,3,4] => ([],4)
=> 1
[-,+,-,-] => [1,2,3,4] => ([],4)
=> 1
[+,-,-,-] => [1,2,3,4] => ([],4)
=> 1
[-,-,-,-] => [1,2,3,4] => ([],4)
=> 1
[+,+,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[-,+,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[+,-,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[-,-,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[+,3,2,+] => [1,3,2,4] => ([(2,3)],4)
=> 2
[-,3,2,+] => [1,3,2,4] => ([(2,3)],4)
=> 2
[+,3,2,-] => [1,3,2,4] => ([(2,3)],4)
=> 2
[-,3,2,-] => [1,3,2,4] => ([(2,3)],4)
=> 2
[+,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[-,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[+,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[-,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[2,1,+,+] => [2,1,3,4] => ([(2,3)],4)
=> 2
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
=> 1
[-] => [1] => ([],1)
=> 1
[+,+] => [1,2] => ([],2)
=> 1
[-,+] => [1,2] => ([],2)
=> 1
[+,-] => [1,2] => ([],2)
=> 1
[-,-] => [1,2] => ([],2)
=> 1
[2,1] => [2,1] => ([(0,1)],2)
=> 2
[+,+,+] => [1,2,3] => ([],3)
=> 1
[-,+,+] => [1,2,3] => ([],3)
=> 1
[+,-,+] => [1,2,3] => ([],3)
=> 1
[+,+,-] => [1,2,3] => ([],3)
=> 1
[-,-,+] => [1,2,3] => ([],3)
=> 1
[-,+,-] => [1,2,3] => ([],3)
=> 1
[+,-,-] => [1,2,3] => ([],3)
=> 1
[-,-,-] => [1,2,3] => ([],3)
=> 1
[+,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[-,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[2,1,+] => [2,1,3] => ([(1,2)],3)
=> 2
[2,1,-] => [2,1,3] => ([(1,2)],3)
=> 2
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[+,+,+,+] => [1,2,3,4] => ([],4)
=> 1
[-,+,+,+] => [1,2,3,4] => ([],4)
=> 1
[+,-,+,+] => [1,2,3,4] => ([],4)
=> 1
[+,+,-,+] => [1,2,3,4] => ([],4)
=> 1
[+,+,+,-] => [1,2,3,4] => ([],4)
=> 1
[-,-,+,+] => [1,2,3,4] => ([],4)
=> 1
[-,+,-,+] => [1,2,3,4] => ([],4)
=> 1
[-,+,+,-] => [1,2,3,4] => ([],4)
=> 1
[+,-,-,+] => [1,2,3,4] => ([],4)
=> 1
[+,-,+,-] => [1,2,3,4] => ([],4)
=> 1
[+,+,-,-] => [1,2,3,4] => ([],4)
=> 1
[-,-,-,+] => [1,2,3,4] => ([],4)
=> 1
[-,-,+,-] => [1,2,3,4] => ([],4)
=> 1
[-,+,-,-] => [1,2,3,4] => ([],4)
=> 1
[+,-,-,-] => [1,2,3,4] => ([],4)
=> 1
[-,-,-,-] => [1,2,3,4] => ([],4)
=> 1
[+,+,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[-,+,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[+,-,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[-,-,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[+,3,2,+] => [1,3,2,4] => ([(2,3)],4)
=> 2
[-,3,2,+] => [1,3,2,4] => ([(2,3)],4)
=> 2
[+,3,2,-] => [1,3,2,4] => ([(2,3)],4)
=> 2
[-,3,2,-] => [1,3,2,4] => ([(2,3)],4)
=> 2
[+,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[-,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[+,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[-,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[2,1,+,+] => [2,1,3,4] => ([(2,3)],4)
=> 2
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => [1]
=> 1
[-] => [1] => [1]
=> 1
[+,+] => [1,2] => [1,1]
=> 1
[-,+] => [1,2] => [1,1]
=> 1
[+,-] => [1,2] => [1,1]
=> 1
[-,-] => [1,2] => [1,1]
=> 1
[2,1] => [2,1] => [2]
=> 2
[+,+,+] => [1,2,3] => [1,1,1]
=> 1
[-,+,+] => [1,2,3] => [1,1,1]
=> 1
[+,-,+] => [1,2,3] => [1,1,1]
=> 1
[+,+,-] => [1,2,3] => [1,1,1]
=> 1
[-,-,+] => [1,2,3] => [1,1,1]
=> 1
[-,+,-] => [1,2,3] => [1,1,1]
=> 1
[+,-,-] => [1,2,3] => [1,1,1]
=> 1
[-,-,-] => [1,2,3] => [1,1,1]
=> 1
[+,3,2] => [1,3,2] => [2,1]
=> 2
[-,3,2] => [1,3,2] => [2,1]
=> 2
[2,1,+] => [2,1,3] => [2,1]
=> 2
[2,1,-] => [2,1,3] => [2,1]
=> 2
[2,3,1] => [2,3,1] => [2,1]
=> 2
[3,1,2] => [3,1,2] => [2,1]
=> 2
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> 1
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> 1
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> 1
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> 1
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> 1
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> 1
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> 1
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> 1
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> 1
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> 1
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> 1
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> 1
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> 1
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> 1
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> 1
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> 1
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> 2
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> 2
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> 2
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> 2
[+,3,4,2] => [1,3,4,2] => [2,1,1]
=> 2
[-,3,4,2] => [1,3,4,2] => [2,1,1]
=> 2
[+,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[-,4,2,3] => [1,4,2,3] => [2,1,1]
=> 2
[2,1,+,+] => [2,1,3,4] => [2,1,1]
=> 2
Description
The largest part of an integer partition.
Matching statistic: St000172
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000172: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
=> 1
[-] => [1] => ([],1)
=> 1
[+,+] => [1,2] => ([],2)
=> 1
[-,+] => [1,2] => ([],2)
=> 1
[+,-] => [1,2] => ([],2)
=> 1
[-,-] => [1,2] => ([],2)
=> 1
[2,1] => [2,1] => ([(0,1)],2)
=> 2
[+,+,+] => [1,2,3] => ([],3)
=> 1
[-,+,+] => [1,2,3] => ([],3)
=> 1
[+,-,+] => [1,2,3] => ([],3)
=> 1
[+,+,-] => [1,2,3] => ([],3)
=> 1
[-,-,+] => [1,2,3] => ([],3)
=> 1
[-,+,-] => [1,2,3] => ([],3)
=> 1
[+,-,-] => [1,2,3] => ([],3)
=> 1
[-,-,-] => [1,2,3] => ([],3)
=> 1
[+,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[-,3,2] => [1,3,2] => ([(1,2)],3)
=> 2
[2,1,+] => [2,1,3] => ([(1,2)],3)
=> 2
[2,1,-] => [2,1,3] => ([(1,2)],3)
=> 2
[2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[+,+,+,+] => [1,2,3,4] => ([],4)
=> 1
[-,+,+,+] => [1,2,3,4] => ([],4)
=> 1
[+,-,+,+] => [1,2,3,4] => ([],4)
=> 1
[+,+,-,+] => [1,2,3,4] => ([],4)
=> 1
[+,+,+,-] => [1,2,3,4] => ([],4)
=> 1
[-,-,+,+] => [1,2,3,4] => ([],4)
=> 1
[-,+,-,+] => [1,2,3,4] => ([],4)
=> 1
[-,+,+,-] => [1,2,3,4] => ([],4)
=> 1
[+,-,-,+] => [1,2,3,4] => ([],4)
=> 1
[+,-,+,-] => [1,2,3,4] => ([],4)
=> 1
[+,+,-,-] => [1,2,3,4] => ([],4)
=> 1
[-,-,-,+] => [1,2,3,4] => ([],4)
=> 1
[-,-,+,-] => [1,2,3,4] => ([],4)
=> 1
[-,+,-,-] => [1,2,3,4] => ([],4)
=> 1
[+,-,-,-] => [1,2,3,4] => ([],4)
=> 1
[-,-,-,-] => [1,2,3,4] => ([],4)
=> 1
[+,+,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[-,+,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[+,-,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[-,-,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 2
[+,3,2,+] => [1,3,2,4] => ([(2,3)],4)
=> 2
[-,3,2,+] => [1,3,2,4] => ([(2,3)],4)
=> 2
[+,3,2,-] => [1,3,2,4] => ([(2,3)],4)
=> 2
[-,3,2,-] => [1,3,2,4] => ([(2,3)],4)
=> 2
[+,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[-,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[+,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[-,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[2,1,+,+] => [2,1,3,4] => ([(2,3)],4)
=> 2
Description
The Grundy number of a graph.
The Grundy number $\Gamma(G)$ is defined to be the largest $k$ such that $G$ admits a greedy $k$-coloring. Any order of the vertices of $G$ induces a greedy coloring by assigning to the $i$-th vertex in this order the smallest positive integer such that the partial coloring remains a proper coloring.
In particular, we have that $\chi(G) \leq \Gamma(G) \leq \Delta(G) + 1$, where $\chi(G)$ is the chromatic number of $G$ ([[St000098]]), and where $\Delta(G)$ is the maximal degree of a vertex of $G$ ([[St000171]]).
Matching statistic: St000298
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => [1] => ([],1)
=> 1
[-] => [1] => ([],1)
=> 1
[+,+] => [1,2] => ([(0,1)],2)
=> 1
[-,+] => [1,2] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => ([(0,1)],2)
=> 1
[-,-] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [2,1] => ([],2)
=> 2
[+,+,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[-,+,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[+,-,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[+,+,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[-,-,+] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[-,+,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[+,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[-,-,-] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[+,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[-,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,1,+] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[2,1,-] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [2,3,1] => ([(1,2)],3)
=> 2
[3,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[+,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[-,+,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[+,-,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[+,+,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[+,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[-,-,+,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[-,+,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[-,+,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[+,-,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[+,-,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[+,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[-,-,-,+] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[-,-,+,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[-,+,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[+,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[-,-,-,-] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[+,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[-,+,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[+,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[-,-,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[+,3,2,+] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[-,3,2,+] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[+,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[-,3,2,-] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[+,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[-,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[+,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[-,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[2,1,+,+] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
The following 359 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000346The number of coarsenings of a partition. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000527The width of the poset. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001330The hat guessing number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000141The maximum drop size of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001358The largest degree of a regular subgraph of a graph. St001587Half of the largest even part of an integer partition. St001644The dimension of a graph. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001928The number of non-overlapping descents in a permutation. St001962The proper pathwidth of a graph. St000013The height of a Dyck path. St000086The number of subgraphs. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000166The depth minus 1 of an ordered tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000288The number of ones in a binary word. St000299The number of nonisomorphic vertex-induced subtrees. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000321The number of integer partitions of n that are dominated by an integer partition. St000328The maximum number of child nodes in a tree. St000343The number of spanning subgraphs of a graph. St000345The number of refinements of a partition. St000363The number of minimal vertex covers of a graph. St000364The exponent of the automorphism group of a graph. St000378The diagonal inversion number of an integer partition. St000381The largest part of an integer composition. St000397The Strahler number of a rooted tree. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000469The distinguishing number of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000479The Ramsey number of a graph. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000920The logarithmic height of a Dyck path. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000935The number of ordered refinements of an integer partition. St000972The composition number of a graph. St000991The number of right-to-left minima of a permutation. St001058The breadth of the ordered tree. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001110The 3-dynamic chromatic number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001486The number of corners of the ribbon associated with an integer composition. St001530The depth of a Dyck path. St001624The breadth of a lattice. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001717The largest size of an interval in a poset. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000080The rank of the poset. St000081The number of edges of a graph. St000094The depth of an ordered tree. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000183The side length of the Durfee square of an integer partition. St000245The number of ascents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000274The number of perfect matchings of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000352The Elizalde-Pak rank of a permutation. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000387The matching number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000632The jump number of the poset. St000651The maximal size of a rise in a permutation. St000703The number of deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001117The game chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001271The competition number of a graph. St001280The number of parts of an integer partition that are at least two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001340The cardinality of a minimal non-edge isolating set of a graph. St001341The number of edges in the center of a graph. St001345The Hamming dimension of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001391The disjunction number of a graph. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001427The number of descents of a signed permutation. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001479The number of bridges of a graph. St001484The number of singletons of an integer partition. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001512The minimum rank of a graph. St001649The length of a longest trail in a graph. St001657The number of twos in an integer partition. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001777The number of weak descents in an integer composition. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001801Half the number of preimage-image pairs of different parity in a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001812The biclique partition number of a graph. St001826The maximal number of leaves on a vertex of a graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001931The weak major index of an integer composition regarded as a word. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001949The rigidity index of a graph. St000485The length of the longest cycle of a permutation. St000668The least common multiple of the parts of the partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000392The length of the longest run of ones in a binary word. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000730The maximal arc length of a set partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000444The length of the maximal rise of a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001568The smallest positive integer that does not appear twice in the partition. St000251The number of nonsingleton blocks of a set partition. St000354The number of recoils of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000640The rank of the largest boolean interval in a poset. St000659The number of rises of length at least 2 of a Dyck path. St000753The Grundy value for the game of Kayles on a binary word. St000829The Ulam distance of a permutation to the identity permutation. St000919The number of maximal left branches of a binary tree. St001114The number of odd descents of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000455The second largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001555The order of a signed permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000464The Schultz index of a connected graph. 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 $eA$. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001118The acyclic chromatic index of a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001281The normalized isoperimetric number of a graph. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. 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. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. 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. St000379The number of Hamiltonian cycles in a graph. 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. St000944The 3-degree of an integer 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001946The number of descents in a parking function. St001570The minimal number of edges to add to make a graph Hamiltonian. St000699The toughness times the least common multiple of 1,. St000806The semiperimeter of the associated bargraph. 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 $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
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