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Your data matches 853 different statistics following compositions of up to 3 maps.
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Matching statistic: St000290
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000290: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 = -1 + 1
[1,1] => 11 => 0 = -1 + 1
[2] => 10 => 1 = 0 + 1
[1,1,1] => 111 => 0 = -1 + 1
[2,1] => 101 => 1 = 0 + 1
[3] => 100 => 1 = 0 + 1
[1,1,1,1] => 1111 => 0 = -1 + 1
[2,1,1] => 1011 => 1 = 0 + 1
[3,1] => 1001 => 1 = 0 + 1
[4] => 1000 => 1 = 0 + 1
[1,1,1,1,1] => 11111 => 0 = -1 + 1
[2,1,1,1] => 10111 => 1 = 0 + 1
[3,1,1] => 10011 => 1 = 0 + 1
[4,1] => 10001 => 1 = 0 + 1
[5] => 10000 => 1 = 0 + 1
[1,1,1,1,1,1] => 111111 => 0 = -1 + 1
[2,1,1,1,1] => 101111 => 1 = 0 + 1
[3,1,1,1] => 100111 => 1 = 0 + 1
[4,1,1] => 100011 => 1 = 0 + 1
[5,1] => 100001 => 1 = 0 + 1
[6] => 100000 => 1 = 0 + 1
Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000291
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 = -1 + 1
[1,1] => 11 => 0 = -1 + 1
[2] => 10 => 1 = 0 + 1
[1,1,1] => 111 => 0 = -1 + 1
[2,1] => 101 => 1 = 0 + 1
[3] => 100 => 1 = 0 + 1
[1,1,1,1] => 1111 => 0 = -1 + 1
[2,1,1] => 1011 => 1 = 0 + 1
[3,1] => 1001 => 1 = 0 + 1
[4] => 1000 => 1 = 0 + 1
[1,1,1,1,1] => 11111 => 0 = -1 + 1
[2,1,1,1] => 10111 => 1 = 0 + 1
[3,1,1] => 10011 => 1 = 0 + 1
[4,1] => 10001 => 1 = 0 + 1
[5] => 10000 => 1 = 0 + 1
[1,1,1,1,1,1] => 111111 => 0 = -1 + 1
[2,1,1,1,1] => 101111 => 1 = 0 + 1
[3,1,1,1] => 100111 => 1 = 0 + 1
[4,1,1] => 100011 => 1 = 0 + 1
[5,1] => 100001 => 1 = 0 + 1
[6] => 100000 => 1 = 0 + 1
Description
The number of descents of a binary word.
Matching statistic: St000480
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0 = -1 + 1
[1,1] => [1,1]
=> 0 = -1 + 1
[2] => [2]
=> 1 = 0 + 1
[1,1,1] => [1,1,1]
=> 0 = -1 + 1
[2,1] => [2,1]
=> 1 = 0 + 1
[3] => [3]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1]
=> 0 = -1 + 1
[2,1,1] => [2,1,1]
=> 1 = 0 + 1
[3,1] => [3,1]
=> 1 = 0 + 1
[4] => [4]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 0 = -1 + 1
[2,1,1,1] => [2,1,1,1]
=> 1 = 0 + 1
[3,1,1] => [3,1,1]
=> 1 = 0 + 1
[4,1] => [4,1]
=> 1 = 0 + 1
[5] => [5]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0 = -1 + 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 1 = 0 + 1
[3,1,1,1] => [3,1,1,1]
=> 1 = 0 + 1
[4,1,1] => [4,1,1]
=> 1 = 0 + 1
[5,1] => [5,1]
=> 1 = 0 + 1
[6] => [6]
=> 1 = 0 + 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
Matching statistic: St000875
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000875: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000875: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 = -1 + 1
[1,1] => 11 => 0 = -1 + 1
[2] => 10 => 1 = 0 + 1
[1,1,1] => 111 => 0 = -1 + 1
[2,1] => 101 => 1 = 0 + 1
[3] => 100 => 1 = 0 + 1
[1,1,1,1] => 1111 => 0 = -1 + 1
[2,1,1] => 1011 => 1 = 0 + 1
[3,1] => 1001 => 1 = 0 + 1
[4] => 1000 => 1 = 0 + 1
[1,1,1,1,1] => 11111 => 0 = -1 + 1
[2,1,1,1] => 10111 => 1 = 0 + 1
[3,1,1] => 10011 => 1 = 0 + 1
[4,1] => 10001 => 1 = 0 + 1
[5] => 10000 => 1 = 0 + 1
[1,1,1,1,1,1] => 111111 => 0 = -1 + 1
[2,1,1,1,1] => 101111 => 1 = 0 + 1
[3,1,1,1] => 100111 => 1 = 0 + 1
[4,1,1] => 100011 => 1 = 0 + 1
[5,1] => 100001 => 1 = 0 + 1
[6] => 100000 => 1 = 0 + 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the semilength of the longest Dyck word in this factorisation.
Matching statistic: St001280
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ 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] => [1]
=> 0 = -1 + 1
[1,1] => [1,1]
=> 0 = -1 + 1
[2] => [2]
=> 1 = 0 + 1
[1,1,1] => [1,1,1]
=> 0 = -1 + 1
[2,1] => [2,1]
=> 1 = 0 + 1
[3] => [3]
=> 1 = 0 + 1
[1,1,1,1] => [1,1,1,1]
=> 0 = -1 + 1
[2,1,1] => [2,1,1]
=> 1 = 0 + 1
[3,1] => [3,1]
=> 1 = 0 + 1
[4] => [4]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 0 = -1 + 1
[2,1,1,1] => [2,1,1,1]
=> 1 = 0 + 1
[3,1,1] => [3,1,1]
=> 1 = 0 + 1
[4,1] => [4,1]
=> 1 = 0 + 1
[5] => [5]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 0 = -1 + 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 1 = 0 + 1
[3,1,1,1] => [3,1,1,1]
=> 1 = 0 + 1
[4,1,1] => [4,1,1]
=> 1 = 0 + 1
[5,1] => [5,1]
=> 1 = 0 + 1
[6] => [6]
=> 1 = 0 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001421
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St001421: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001421: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 = -1 + 1
[1,1] => 11 => 0 = -1 + 1
[2] => 10 => 1 = 0 + 1
[1,1,1] => 111 => 0 = -1 + 1
[2,1] => 101 => 1 = 0 + 1
[3] => 100 => 1 = 0 + 1
[1,1,1,1] => 1111 => 0 = -1 + 1
[2,1,1] => 1011 => 1 = 0 + 1
[3,1] => 1001 => 1 = 0 + 1
[4] => 1000 => 1 = 0 + 1
[1,1,1,1,1] => 11111 => 0 = -1 + 1
[2,1,1,1] => 10111 => 1 = 0 + 1
[3,1,1] => 10011 => 1 = 0 + 1
[4,1] => 10001 => 1 = 0 + 1
[5] => 10000 => 1 = 0 + 1
[1,1,1,1,1,1] => 111111 => 0 = -1 + 1
[2,1,1,1,1] => 101111 => 1 = 0 + 1
[3,1,1,1] => 100111 => 1 = 0 + 1
[4,1,1] => 100011 => 1 = 0 + 1
[5,1] => 100001 => 1 = 0 + 1
[6] => 100000 => 1 = 0 + 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Matching statistic: St001485
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St001485: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001485: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 = -1 + 1
[1,1] => 11 => 0 = -1 + 1
[2] => 10 => 1 = 0 + 1
[1,1,1] => 111 => 0 = -1 + 1
[2,1] => 101 => 1 = 0 + 1
[3] => 100 => 1 = 0 + 1
[1,1,1,1] => 1111 => 0 = -1 + 1
[2,1,1] => 1011 => 1 = 0 + 1
[3,1] => 1001 => 1 = 0 + 1
[4] => 1000 => 1 = 0 + 1
[1,1,1,1,1] => 11111 => 0 = -1 + 1
[2,1,1,1] => 10111 => 1 = 0 + 1
[3,1,1] => 10011 => 1 = 0 + 1
[4,1] => 10001 => 1 = 0 + 1
[5] => 10000 => 1 = 0 + 1
[1,1,1,1,1,1] => 111111 => 0 = -1 + 1
[2,1,1,1,1] => 101111 => 1 = 0 + 1
[3,1,1,1] => 100111 => 1 = 0 + 1
[4,1,1] => 100011 => 1 = 0 + 1
[5,1] => 100001 => 1 = 0 + 1
[6] => 100000 => 1 = 0 + 1
Description
The modular major index of a binary word.
This is [[St000290]] modulo the length of the word.
Matching statistic: St001642
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001642: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001642: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1 = -1 + 2
[1,1] => ([(0,1)],2)
=> 1 = -1 + 2
[2] => ([],2)
=> 2 = 0 + 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = -1 + 2
[2,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[3] => ([],3)
=> 2 = 0 + 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = -1 + 2
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4] => ([],4)
=> 2 = 0 + 2
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = -1 + 2
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[5] => ([],5)
=> 2 = 0 + 2
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = -1 + 2
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[6] => ([],6)
=> 2 = 0 + 2
Description
The Prague dimension of a graph.
This is the least number of complete graphs such that the graph is an induced subgraph of their (categorical) product.
Put differently, this is the least number $n$ such that the graph can be embedded into $\mathbb N^n$, where two points are connected by an edge if and only if they differ in all coordinates.
Matching statistic: St000455
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Values
[1] => ([],1)
=> ([(0,1)],2)
=> -1
[1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> -1
[2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> -1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[5] => ([],5)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> -1
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[6] => ([],6)
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000004
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0 = -1 + 1
[1,1] => [1,0,1,0]
=> [1,2] => 0 = -1 + 1
[2] => [1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = -1 + 1
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [3,1,2] => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = -1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = -1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0 = -1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1 = 0 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6] => 1 = 0 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,2,3,5,6] => 1 = 0 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 1 = 0 + 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 1 = 0 + 1
Description
The major index of a permutation.
This is the sum of the positions of its descents,
$$\operatorname{maj}(\sigma) = \sum_{\sigma(i) > \sigma(i+1)} i.$$
Its generating function is $[n]_q! = [1]_q \cdot [2]_q \dots [n]_q$ for $[k]_q = 1 + q + q^2 + \dots q^{k-1}$.
A statistic equidistributed with the major index is called '''Mahonian statistic'''.
The following 843 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. 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. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000154The sum of the descent bottoms of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000156The Denert index of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000347The inversion sum of a binary word. St000352The Elizalde-Pak rank of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000390The number of runs of ones in a binary word. St000481The number of upper covers of a partition in dominance order. St000535The rank-width of a graph. St000651The maximal size of a rise in a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000691The number of changes of a binary word. St000761The number of ascents in an integer composition. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000884The number of isolated descents of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001090The number of pop-stack-sorts needed to sort a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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 St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. 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. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001375The pancake length of a permutation. St001393The induced matching number of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001512The minimum rank of a graph. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001874Lusztig's a-function for the symmetric group. St001928The number of non-overlapping descents in a permutation. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000058The order of a permutation. St000259The diameter of a connected graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000346The number of coarsenings of a partition. St000388The number of orbits of vertices of a graph under automorphisms. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000758The length of the longest staircase fitting into an integer composition. St000764The number of strong records in an integer composition. St000903The number of different parts of an integer composition. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001093The detour number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001261The Castelnuovo-Mumford regularity of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001674The number of vertices of the largest induced star graph in the graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000008The major index of the composition. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000055The inversion sum of a permutation. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000133The "bounce" of a permutation. St000141The maximum drop size of a permutation. St000142The number of even parts of a partition. St000147The largest part of an integer partition. 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. St000183The side length of the Durfee square of an integer partition. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000224The sorting index of a permutation. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000238The number of indices that are not small weak excedances. 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. St000260The radius of a connected graph. St000272The treewidth of a graph. St000295The length of the border of a binary word. St000297The number of leading ones in a binary word. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000330The (standard) major index of a standard tableau. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000336The leg major index of a standard tableau. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000339The maf index of a permutation. St000348The non-inversion sum of a binary word. St000356The number of occurrences of the pattern 13-2. St000362The size of a minimal vertex cover of a graph. St000378The diagonal inversion number of an integer partition. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000446The disorder of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000534The number of 2-rises of a permutation. St000536The pathwidth of a graph. St000546The number of global descents of a permutation. St000552The number of cut vertices of a graph. St000628The balance of a binary word. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000672The number of minimal elements in Bruhat order not less than the permutation. St000682The Grundy value of Welter's game on a binary word. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000703The number of deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000742The number of big ascents of a permutation after prepending zero. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000783The side length of the largest staircase partition fitting into a partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000868The aid statistic in the sense of Shareshian-Wachs. St000897The number of different multiplicities of parts of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001092The number of distinct even parts of a partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001153The number of blocks with even minimum in a set partition. St001161The major index north count of a Dyck path. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001274The number of indecomposable injective modules with projective dimension equal to two. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001358The largest degree of a regular subgraph of a graph. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001484The number of singletons of an integer partition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001587Half of the largest even part of an integer partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001657The number of twos in an integer partition. St001671Haglund's hag of a permutation. St001673The degree of asymmetry of an integer composition. St001689The number of celebrities in a graph. St001691The number of kings in a graph. St001712The number of natural descents of a standard Young tableau. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001812The biclique partition number 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). St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001961The sum of the greatest common divisors of all pairs of parts. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000040The number of regions of the inversion arrangement of a permutation. St000054The first entry of the permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000084The number of subtrees. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000109The number of elements less than or equal to the given element in Bruhat order. St000110The number of permutations less than or equal to a permutation in left weak order. St000153The number of adjacent cycles of a permutation. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000258The burning number of a graph. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. 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. St000326The position of the first one in a binary word after appending a 1 at the end. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000345The number of refinements of a partition. St000396The register function (or Horton-Strahler number) of a binary tree. St000397The Strahler number of a rooted tree. St000402Half the size of the symmetry class of a permutation. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000527The width of the poset. St000568The hook number of a binary tree. St000638The number of up-down runs of a permutation. St000701The protection number of a binary tree. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000759The smallest missing part in an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000822The Hadwiger number of the graph. St000843The decomposition number of a perfect matching. St000847The number of standard Young tableaux whose descent set is the binary word. St000862The number of parts of the shifted shape of a permutation. St000883The number of longest increasing subsequences of a permutation. St000918The 2-limited packing 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. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001315The dissociation 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. St001330The hat guessing number of a graph. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001494The Alon-Tarsi number of a graph. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001670The connected partition number of a graph. St001732The number of peaks visible from the left. St001746The coalition number of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001884The number of borders of a binary word. St001963The tree-depth of a graph. St000439The position of the first down step of a Dyck path. St000891The number of distinct diagonal sums of a permutation matrix. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001486The number of corners of the ribbon associated with an integer composition. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000653The last descent of a permutation. St000730The maximal arc length of a set partition. St000794The mak of a permutation. 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. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000485The length of the longest cycle of a permutation. St000990The first ascent of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000288The number of ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000472The sum of the ascent bottoms of a permutation. St000475The number of parts equal to 1 in a partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000548The number of different non-empty partial sums of an integer partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000640The rank of the largest boolean interval in a poset. St000646The number of big ascents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000976The sum of the positions of double up-steps of a Dyck path. St000984The number of boxes below precisely one peak. St000989The number of final rises of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001480The number of simple summands of the module J^2/J^3. St001592The maximal number of simple paths between any two different vertices of a graph. St000061The number of nodes on the left branch of a binary tree. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000619The number of cyclic descents of a permutation. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000702The number of weak deficiencies of a permutation. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000844The size of the largest block in the direct sum decomposition of a permutation. St000925The number of topologically connected components of a set partition. St000933The number of multipartitions of sizes given by an integer partition. St001052The length of the exterior of a permutation. St001062The maximal size of a block of a set partition. St001096The size of the overlap set of a permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001346The number of parking functions that give the same permutation. St001352The number of internal nodes in the modular decomposition of a graph. St001500The global dimension of magnitude 1 Nakayama algebras. St001531Number of partial orders contained in the poset determined by the Dyck path. St001735The number of permutations with the same set of runs. St001959The product of the heights of the peaks of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000023The number of inner peaks of a permutation. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000358The number of occurrences of the pattern 31-2. St000486The number of cycles of length at least 3 of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001394The genus of a permutation. St001727The number of invisible inversions of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the 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. St000699The toughness times the least common multiple of 1,. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. 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 $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001734The lettericity of a graph. 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. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000376The bounce deficit of a Dyck path. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001175The size of a partition minus the hook length of the base cell. 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. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001281The normalized isoperimetric number of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000242The number of indices that are not cyclical small weak excedances. St000317The cycle descent number of a permutation. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000461The rix statistic of a permutation. St000490The intertwining number of a set partition. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000618The number of self-evacuating tableaux of given shape. St000624The normalized sum of the minimal distances to a greater element. St000667The greatest common divisor of the parts of the partition. St000717The number of ordinal summands of a poset. St000731The number of double exceedences of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St000840The number of closers smaller than the largest opener in a perfect matching. St000873The aix statistic of a permutation. St000906The length of the shortest maximal chain in a poset. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001095The number of non-isomorphic posets with precisely one further covering relation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001684The reduced word complexity of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001728The number of invisible descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001914The size of the orbit 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. 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. St000060The greater neighbor of the maximum. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000420The number of Dyck paths that are weakly above a Dyck path. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000988The orbit size of a permutation under Foata's bijection. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001568The smallest positive integer that does not appear twice in the partition. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001754The number of tolerances of a finite lattice. St001808The box weight or horizontal decoration of a Dyck path. St000264The girth of a graph, which is not a tree. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000806The semiperimeter of the associated bargraph. St000145The Dyson rank of a partition. St000327The number of cover relations in a poset. St001668The number of points of the poset minus the width of the poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001569The maximal modular displacement of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001890The maximum magnitude of the Möbius function of a poset. St001948The number of augmented double ascents of a permutation. St001555The order of a signed permutation. St000478Another weight of a partition according to Alladi. 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. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000311The number of vertices of odd degree in a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. 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. 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. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001323The independence gap of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001395The number of strictly unfriendly partitions of a graph. St001479The number of bridges of a graph. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. 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. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001826The maximal number of leaves on a vertex of a graph. St000284The Plancherel distribution on integer partitions. St000287The number of connected components of a graph. St000553The number of blocks of a graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000916The packing number of a graph. St000917The open packing number of a graph. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. 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. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001118The acyclic chromatic index of a graph. 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 $eA$. St000117The number of centered tunnels of a Dyck path. St000296The length of the symmetric border of a binary word. St000369The dinv deficit of a Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000629The defect of a binary word. St000661The number of rises of length 3 of a Dyck path. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000934The 2-degree of an integer 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. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001025Number of simple modules with projective dimension 4 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. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001371The length of the longest Yamanouchi prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001570The minimal number of edges to add to make a graph Hamiltonian. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001695The natural comajor index of a standard Young tableau. 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. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001910The height of the middle non-run of a Dyck path. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000456The monochromatic index of a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000466The Gutman (or modified Schultz) index of a connected graph. St000474Dyson's crank of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000675The number of centered multitunnels of a Dyck path. 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. St000762The sum of the positions of the weak records of an integer composition. 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. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000913The number of ways to refine the partition into singletons. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. 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 $K[x]/(x^n)$. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001256Number of simple reflexive modules that are 2-stable reflexive. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the 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. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation 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. St001557The number of inversions of the second entry of a permutation. St001561The value of the elementary symmetric function evaluated at 1. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. 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. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001763The Hurwitz number of an integer partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001838The number of nonempty primitive factors of a binary word. 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. St001960The number of descents of a permutation minus one if its first entry is not one. St000464The Schultz index of a connected graph. 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. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001462The number of factors of a standard tableaux under concatenation. St001545The second Elser number of a connected graph. St001645The pebbling number of a connected graph. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two 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. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St001115The number of even descents of a permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000307The number of rowmotion orbits of a poset. 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. St001490The number of connected components of a skew partition. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001060The distinguishing index of a graph. St001651The Frankl number of a lattice. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000302The determinant of the distance matrix of a connected graph. St000467The hyper-Wiener index of a connected graph. St000649The number of 3-excedences of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St000736The last entry in the first row of a semistandard tableau. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion.
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