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Your data matches 311 different statistics following compositions of up to 3 maps.
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Matching statistic: St000003
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000003: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1 = 0 + 1
[1,2] => [2]
=> 1 = 0 + 1
[2,1] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [3]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [1,1,1]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> 3 = 2 + 1
[1,3,4,2] => [3,1]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[2,1,3,4] => [3,1]
=> 3 = 2 + 1
[2,3,4,1] => [3,1]
=> 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1]
=> 3 = 2 + 1
[4,1,2,3] => [3,1]
=> 3 = 2 + 1
[4,2,1,3] => [2,1,1]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,1]
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> 1 = 0 + 1
[5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[6,5,4,3,2,1] => [1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5,6,7] => [7]
=> 1 = 0 + 1
[7,6,5,4,3,2,1] => [1,1,1,1,1,1,1]
=> 1 = 0 + 1
Description
The number of [[/StandardTableaux|standard Young tableaux]] of the partition.
Matching statistic: St000277
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000277: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 = 0 + 1
[1,2] => [2] => 1 = 0 + 1
[2,1] => [1,1] => 1 = 0 + 1
[1,2,3] => [3] => 1 = 0 + 1
[1,3,2] => [2,1] => 2 = 1 + 1
[2,1,3] => [1,2] => 2 = 1 + 1
[2,3,1] => [2,1] => 2 = 1 + 1
[3,1,2] => [1,2] => 2 = 1 + 1
[3,2,1] => [1,1,1] => 1 = 0 + 1
[1,2,3,4] => [4] => 1 = 0 + 1
[1,2,4,3] => [3,1] => 3 = 2 + 1
[1,3,4,2] => [3,1] => 3 = 2 + 1
[1,4,3,2] => [2,1,1] => 3 = 2 + 1
[2,1,3,4] => [1,3] => 3 = 2 + 1
[2,3,4,1] => [3,1] => 3 = 2 + 1
[3,2,1,4] => [1,1,2] => 3 = 2 + 1
[3,4,2,1] => [2,1,1] => 3 = 2 + 1
[4,1,2,3] => [1,3] => 3 = 2 + 1
[4,2,1,3] => [1,1,2] => 3 = 2 + 1
[4,3,1,2] => [1,1,2] => 3 = 2 + 1
[4,3,2,1] => [1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5] => [5] => 1 = 0 + 1
[5,4,3,2,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5,6] => [6] => 1 = 0 + 1
[6,5,4,3,2,1] => [1,1,1,1,1,1] => 1 = 0 + 1
[1,2,3,4,5,6,7] => [7] => 1 = 0 + 1
[7,6,5,4,3,2,1] => [1,1,1,1,1,1,1] => 1 = 0 + 1
Description
The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' [[Mp00071]] from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals
$$
\sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell},
$$
where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]
Matching statistic: St001780
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001780: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1 = 0 + 1
[1,2] => [2]
=> 1 = 0 + 1
[2,1] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [3]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [1,1,1]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> 3 = 2 + 1
[1,3,4,2] => [3,1]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[2,1,3,4] => [3,1]
=> 3 = 2 + 1
[2,3,4,1] => [3,1]
=> 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1]
=> 3 = 2 + 1
[4,1,2,3] => [3,1]
=> 3 = 2 + 1
[4,2,1,3] => [2,1,1]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,1]
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> 1 = 0 + 1
[5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[6,5,4,3,2,1] => [1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5,6,7] => [7]
=> 1 = 0 + 1
[7,6,5,4,3,2,1] => [1,1,1,1,1,1,1]
=> 1 = 0 + 1
Description
The order of promotion on the set of standard tableaux of given shape.
Matching statistic: St001908
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001908: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001908: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1 = 0 + 1
[1,2] => [2]
=> 1 = 0 + 1
[2,1] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [3]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> 2 = 1 + 1
[2,1,3] => [2,1]
=> 2 = 1 + 1
[2,3,1] => [2,1]
=> 2 = 1 + 1
[3,1,2] => [2,1]
=> 2 = 1 + 1
[3,2,1] => [1,1,1]
=> 1 = 0 + 1
[1,2,3,4] => [4]
=> 1 = 0 + 1
[1,2,4,3] => [3,1]
=> 3 = 2 + 1
[1,3,4,2] => [3,1]
=> 3 = 2 + 1
[1,4,3,2] => [2,1,1]
=> 3 = 2 + 1
[2,1,3,4] => [3,1]
=> 3 = 2 + 1
[2,3,4,1] => [3,1]
=> 3 = 2 + 1
[3,2,1,4] => [2,1,1]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,1]
=> 3 = 2 + 1
[4,1,2,3] => [3,1]
=> 3 = 2 + 1
[4,2,1,3] => [2,1,1]
=> 3 = 2 + 1
[4,3,1,2] => [2,1,1]
=> 3 = 2 + 1
[4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5]
=> 1 = 0 + 1
[5,4,3,2,1] => [1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5,6] => [6]
=> 1 = 0 + 1
[6,5,4,3,2,1] => [1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,3,4,5,6,7] => [7]
=> 1 = 0 + 1
[7,6,5,4,3,2,1] => [1,1,1,1,1,1,1]
=> 1 = 0 + 1
Description
The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition.
For example, there are eight tableaux of shape $[3,2,1]$ with maximal entry $3$, but two of them have the same weight.
Matching statistic: St000057
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 0
[1,2] => [1,1]
=> [[1],[2]]
=> 0
[2,1] => [2]
=> [[1,2]]
=> 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> 1
[2,3,1] => [2,1]
=> [[1,2],[3]]
=> 1
[3,1,2] => [2,1]
=> [[1,2],[3]]
=> 1
[3,2,1] => [3]
=> [[1,2,3]]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 2
[2,1,3,4] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,3,4,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[3,2,1,4] => [3,1]
=> [[1,2,3],[4]]
=> 2
[3,4,2,1] => [3,1]
=> [[1,2,3],[4]]
=> 2
[4,1,2,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[4,2,1,3] => [3,1]
=> [[1,2,3],[4]]
=> 2
[4,3,1,2] => [3,1]
=> [[1,2,3],[4]]
=> 2
[4,3,2,1] => [4]
=> [[1,2,3,4]]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,4,3,2,1] => [5]
=> [[1,2,3,4,5]]
=> 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[6,5,4,3,2,1] => [6]
=> [[1,2,3,4,5,6]]
=> 0
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[7,6,5,4,3,2,1] => [7]
=> [[1,2,3,4,5,6,7]]
=> 0
Description
The Shynar inversion number of a standard tableau.
Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.
Matching statistic: St000387
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
Description
The matching number of a graph.
For a graph $G$, this is defined as the maximal size of a '''matching''' or '''independent edge set''' (a set of edges without common vertices) contained in $G$.
Matching statistic: St000682
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000682: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000682: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 0
[1,2] => [1,1]
=> 110 => 0
[2,1] => [2]
=> 100 => 0
[1,2,3] => [1,1,1]
=> 1110 => 0
[1,3,2] => [2,1]
=> 1010 => 1
[2,1,3] => [2,1]
=> 1010 => 1
[2,3,1] => [2,1]
=> 1010 => 1
[3,1,2] => [2,1]
=> 1010 => 1
[3,2,1] => [3]
=> 1000 => 0
[1,2,3,4] => [1,1,1,1]
=> 11110 => 0
[1,2,4,3] => [2,1,1]
=> 10110 => 2
[1,3,4,2] => [2,1,1]
=> 10110 => 2
[1,4,3,2] => [3,1]
=> 10010 => 2
[2,1,3,4] => [2,1,1]
=> 10110 => 2
[2,3,4,1] => [2,1,1]
=> 10110 => 2
[3,2,1,4] => [3,1]
=> 10010 => 2
[3,4,2,1] => [3,1]
=> 10010 => 2
[4,1,2,3] => [2,1,1]
=> 10110 => 2
[4,2,1,3] => [3,1]
=> 10010 => 2
[4,3,1,2] => [3,1]
=> 10010 => 2
[4,3,2,1] => [4]
=> 10000 => 0
[1,2,3,4,5] => [1,1,1,1,1]
=> 111110 => 0
[5,4,3,2,1] => [5]
=> 100000 => 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 1111110 => 0
[6,5,4,3,2,1] => [6]
=> 1000000 => 0
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> 11111110 => 0
[7,6,5,4,3,2,1] => [7]
=> 10000000 => 0
Description
The Grundy value of Welter's game on a binary word.
Two players take turns moving a $1$ to the left. The loosing positions are the words $1\dots 10\dots 0$.
Matching statistic: St000985
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> 0
[1,2] => ([(0,1)],2)
=> ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
[7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
Description
The number of positive eigenvalues of the adjacency matrix of the graph.
Matching statistic: St001633
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00209: Permutations —pattern poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => ([(0,1)],2)
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[1,3,4,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2
[1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[2,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[4,1,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[4,2,1,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2
[4,3,1,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2
[4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001698
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 0
[1,2] => [1,1]
=> [[1],[2]]
=> 0
[2,1] => [2]
=> [[1,2]]
=> 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,3,2] => [2,1]
=> [[1,3],[2]]
=> 1
[2,1,3] => [2,1]
=> [[1,3],[2]]
=> 1
[2,3,1] => [2,1]
=> [[1,3],[2]]
=> 1
[3,1,2] => [2,1]
=> [[1,3],[2]]
=> 1
[3,2,1] => [3]
=> [[1,2,3]]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[1,3,4,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[1,4,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 2
[2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[2,3,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[3,2,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 2
[3,4,2,1] => [3,1]
=> [[1,3,4],[2]]
=> 2
[4,1,2,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[4,2,1,3] => [3,1]
=> [[1,3,4],[2]]
=> 2
[4,3,1,2] => [3,1]
=> [[1,3,4],[2]]
=> 2
[4,3,2,1] => [4]
=> [[1,2,3,4]]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[5,4,3,2,1] => [5]
=> [[1,2,3,4,5]]
=> 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[6,5,4,3,2,1] => [6]
=> [[1,2,3,4,5,6]]
=> 0
[1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[7,6,5,4,3,2,1] => [7]
=> [[1,2,3,4,5,6,7]]
=> 0
Description
The comajor index of a standard tableau minus the weighted size of its shape.
The following 301 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001699The major index of a standard tableau minus the weighted size of its shape. St000047The number of standard immaculate tableaux of a given shape. St000172The Grundy number of a graph. St001102The number of words with multiplicities of the letters given by the composition, avoiding the consecutive pattern 132. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001581The achromatic number of a graph. St001595The number of standard Young tableaux of the skew partition. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001963The tree-depth of a graph. St000089The absolute variation of a composition. St000185The weighted size of a partition. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000293The number of inversions of a binary word. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000496The rcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St001176The size of a partition minus its first part. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001280The number of parts of an integer partition that are at least two. St001584The area statistic between a Dyck path and its bounce path. St001695The natural comajor index of a standard Young tableau. St001961The sum of the greatest common divisors of all pairs of parts. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000670The reversal length of a permutation. 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. St001304The number of maximally independent sets of vertices of a graph. St001725The harmonious chromatic number of a graph. 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. St000075The orbit size of a standard tableau under promotion. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000369The dinv deficit of a Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001910The height of the middle non-run of a Dyck path. St000071The number of maximal chains in a poset. St000909The number of maximal chains of maximal size in a poset. St001312Number of parabolic noncrossing partitions indexed by the composition. St001415The length of the longest palindromic prefix of a binary word. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000290The major index of a binary word. St000355The number of occurrences of the pattern 21-3. St000376The bounce deficit of a Dyck path. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000491The number of inversions of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000747A variant of the major index of a set partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001485The modular major index of a binary word. St000014The number of parking functions supported by a Dyck path. St000048The multinomial of the parts of a partition. St000100The number of linear extensions of a poset. St000452The number of distinct eigenvalues of a graph. St000548The number of different non-empty partial sums of an integer partition. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St000530The number of permutations with the same descent word as the given permutation. St000222The number of alignments in the permutation. St000171The degree of the graph. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001812The biclique partition number of a graph. St000001The number of reduced words for a permutation. St000363The number of minimal vertex covers of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St001110The 3-dynamic chromatic number of a graph. St001268The size of the largest ordinal summand in the poset. St001367The smallest number which does not occur as degree of a vertex in a graph. St001779The order of promotion on the set of linear extensions of a poset. St000058The order of a permutation. 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. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000567The sum of the products of all pairs of parts. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St000209Maximum difference of elements in cycles. St000420The number of Dyck paths that are weakly above a Dyck path. St000456The monochromatic index of a connected graph. St000494The number of inversions of distance at most 3 of a permutation. St000789The number of crossing-similar perfect matchings of a perfect matching. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). 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. St001760The number of prefix or suffix reversals needed to sort a permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000141The maximum drop size of a permutation. St000485The length of the longest cycle of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001438The number of missing boxes of a skew partition. St001535The number of cyclic alignments of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000029The depth of a permutation. St000216The absolute length of a permutation. 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. St001480The number of simple summands of the module J^2/J^3. St000219The number of occurrences of the pattern 231 in a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000045The number of linear extensions of a binary tree. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001727The number of invisible inversions of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000259The diameter of a connected graph. St001118The acyclic chromatic index of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001875The number of simple modules with projective dimension at most 1. St000552The number of cut vertices of a graph. St000768The number of peaks in an integer composition. 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. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001470The cyclic holeyness of a permutation. St001793The difference between the clique number and the chromatic number of a graph. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000785The number of distinct colouring schemes of a graph. St001282The number of graphs with the same chromatic polynomial. St001333The cardinality of a minimal edge-isolating set of a graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000098The chromatic number of a graph. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000636The hull number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001691The number of kings in a 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. St001964The interval resolution global dimension of a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000095The number of triangles of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001607The number of coloured graphs 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. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001488The number of corners of a skew partition. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. St000264The girth of a graph, which is not a tree. 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. St000181The number of connected components of the Hasse diagram for the poset. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000741The Colin de Verdière graph invariant. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001890The maximum magnitude of the Möbius function of a poset. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. 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. St001568The smallest positive integer that does not appear twice in the partition. St000102The charge of a semistandard tableau. St000101The cocharge of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000315The number of isolated vertices of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001625The Möbius invariant of a lattice. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001783The number of odd automorphisms of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001569The maximal modular displacement of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001828The Euler characteristic of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St000822The Hadwiger number of the graph. St001734The lettericity of a graph.
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