Identifier
-
Mp00067:
Permutations
—Foata bijection⟶
Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤ
Values
[1] => [1] => [1] => ([],1) => 0
[1,2] => [1,2] => [2] => ([],2) => 0
[2,1] => [2,1] => [1,1] => ([(0,1)],2) => 1
[1,2,3] => [1,2,3] => [3] => ([],3) => 0
[1,3,2] => [3,1,2] => [1,2] => ([(1,2)],3) => 1
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3) => 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4) => 0
[1,2,4,3] => [4,1,2,3] => [1,3] => ([(2,3)],4) => 1
[1,3,2,4] => [3,1,2,4] => [1,3] => ([(2,3)],4) => 1
[1,4,3,2] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 2
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4) => 1
[2,1,4,3] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5) => 0
[1,2,3,5,4] => [5,1,2,3,4] => [1,4] => ([(3,4)],5) => 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,4] => ([(3,4)],5) => 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[1,3,2,4,5] => [3,1,2,4,5] => [1,4] => ([(3,4)],5) => 1
[1,3,2,5,4] => [5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[1,4,3,2,5] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5) => 1
[2,1,3,5,4] => [5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[2,1,4,3,5] => [4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[2,1,5,4,3] => [5,4,2,1,3] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[2,3,4,5,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,2,1,4,5] => [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 2
[3,2,1,5,4] => [5,3,2,1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[3,4,5,1,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[3,4,5,2,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,5,1,2,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[4,5,2,3,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[4,5,3,1,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[5,1,2,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 2
[5,2,3,4,1] => [2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[5,3,4,1,2] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[5,4,1,2,3] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 3
[5,4,3,2,1] => [5,4,3,2,1] => [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) => 4
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6] => ([],6) => 0
[1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,5] => ([(4,5)],6) => 1
[1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,5] => ([(4,5)],6) => 1
[1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,5] => ([(4,5)],6) => 1
[1,2,4,3,6,5] => [6,4,1,2,3,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[1,2,5,4,3,6] => [5,4,1,2,3,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[1,2,6,5,4,3] => [6,5,4,1,2,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,3,2,4,5,6] => [3,1,2,4,5,6] => [1,5] => ([(4,5)],6) => 1
[1,3,2,4,6,5] => [6,3,1,2,4,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[1,3,2,5,4,6] => [5,3,1,2,4,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[1,3,2,6,5,4] => [6,5,3,1,2,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,4,3,2,5,6] => [4,3,1,2,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[1,4,3,2,6,5] => [6,4,3,1,2,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,5,4,3,2,6] => [5,4,3,1,2,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,5,6,3,2,4] => [1,5,6,3,2,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,6,2,3,4,5] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[1,6,5,2,3,4] => [1,2,6,5,3,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[1,6,5,4,3,2] => [6,5,4,3,1,2] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,5] => ([(4,5)],6) => 1
[2,1,3,4,6,5] => [6,2,1,3,4,5] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[2,1,3,5,4,6] => [5,2,1,3,4,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[2,1,3,6,5,4] => [6,5,2,1,3,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,1,4,3,5,6] => [4,2,1,3,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[2,1,4,3,6,5] => [6,4,2,1,3,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,1,5,4,3,6] => [5,4,2,1,3,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[2,1,6,5,4,3] => [6,5,4,2,1,3] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[2,6,3,4,1,5] => [2,3,6,4,1,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 2
[3,2,1,4,6,5] => [6,3,2,1,4,5] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,1,5,4,6] => [5,3,2,1,4,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,2,1,6,5,4] => [6,5,3,2,1,4] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[3,4,5,1,2,6] => [1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,4,5,1,6,2] => [3,4,5,6,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[3,4,5,2,1,6] => [3,4,5,2,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[3,6,4,1,2,5] => [1,3,6,4,2,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,3,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,3,2,1,6,5] => [6,4,3,2,1,5] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[4,5,1,2,3,6] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[4,5,1,6,2,3] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[4,5,1,6,3,2] => [4,5,6,3,1,2] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,5,2,1,6,3] => [4,5,6,2,1,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,5,2,3,1,6] => [2,4,5,3,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,5,3,1,2,6] => [1,4,5,3,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[4,5,6,2,1,3] => [2,4,5,6,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[5,1,2,3,4,6] => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[5,1,6,2,3,4] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[5,1,6,4,2,3] => [1,5,6,4,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,2,3,4,1,6] => [2,3,5,4,1,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,2,6,3,1,4] => [2,5,6,3,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,3,4,1,2,6] => [1,3,5,4,2,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,4,1,2,3,6] => [1,2,5,4,3,6] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4
[5,6,2,3,1,4] => [2,3,5,6,1,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[5,6,3,1,2,4] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 2
[5,6,3,1,4,2] => [3,5,6,4,1,2] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,6,3,2,1,4] => [3,5,6,2,1,4] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[5,6,4,2,1,3] => [2,5,6,4,1,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
[6,1,4,2,3,5] => [1,2,6,4,3,5] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 3
>>> Load all 282 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
- If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
- If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
- $1$
- $|1|4 \to 14$
- $|14|2 \to 412$
- $|4|1|2|5 \to 4125$
- $|4|125|3 \to 45123.$
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
Map
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!