Identifier
Values
=>
Cc0020;cc-rep
[+]=>[1]=>[1]=>([],1)=>0 [-]=>[1]=>[1]=>([],1)=>0 [-,+]=>[2,1]=>[2,1]=>([(0,1)],2)=>1 [2,1]=>[2,1]=>[2,1]=>([(0,1)],2)=>1 [-,+,+]=>[2,3,1]=>[2,3,1]=>([(0,2),(1,2)],3)=>1 [+,-,+]=>[1,3,2]=>[3,1,2]=>([(0,2),(1,2)],3)=>1 [+,3,2]=>[1,3,2]=>[3,1,2]=>([(0,2),(1,2)],3)=>1 [2,1,+]=>[2,3,1]=>[2,3,1]=>([(0,2),(1,2)],3)=>1 [3,1,2]=>[2,3,1]=>[2,3,1]=>([(0,2),(1,2)],3)=>1 [3,+,1]=>[2,3,1]=>[2,3,1]=>([(0,2),(1,2)],3)=>1 [-,+,+,+]=>[2,3,4,1]=>[2,3,4,1]=>([(0,3),(1,3),(2,3)],4)=>1 [+,-,+,+]=>[1,3,4,2]=>[3,1,4,2]=>([(0,3),(1,2),(2,3)],4)=>2 [+,+,-,+]=>[1,2,4,3]=>[4,1,2,3]=>([(0,3),(1,3),(2,3)],4)=>1 [+,+,4,3]=>[1,2,4,3]=>[4,1,2,3]=>([(0,3),(1,3),(2,3)],4)=>1 [+,3,2,+]=>[1,3,4,2]=>[3,1,4,2]=>([(0,3),(1,2),(2,3)],4)=>2 [+,4,2,3]=>[1,3,4,2]=>[3,1,4,2]=>([(0,3),(1,2),(2,3)],4)=>2 [+,4,+,2]=>[1,3,4,2]=>[3,1,4,2]=>([(0,3),(1,2),(2,3)],4)=>2 [2,1,+,+]=>[2,3,4,1]=>[2,3,4,1]=>([(0,3),(1,3),(2,3)],4)=>1 [3,1,2,+]=>[2,3,4,1]=>[2,3,4,1]=>([(0,3),(1,3),(2,3)],4)=>1 [3,+,1,+]=>[2,3,4,1]=>[2,3,4,1]=>([(0,3),(1,3),(2,3)],4)=>1 [4,1,2,3]=>[2,3,4,1]=>[2,3,4,1]=>([(0,3),(1,3),(2,3)],4)=>1 [4,1,+,2]=>[2,3,4,1]=>[2,3,4,1]=>([(0,3),(1,3),(2,3)],4)=>1 [4,+,1,3]=>[2,3,4,1]=>[2,3,4,1]=>([(0,3),(1,3),(2,3)],4)=>1 [4,+,+,1]=>[2,3,4,1]=>[2,3,4,1]=>([(0,3),(1,3),(2,3)],4)=>1 [-,+,+,+,+]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [+,-,+,+,+]=>[1,3,4,5,2]=>[3,1,4,5,2]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,+,-,+,+]=>[1,2,4,5,3]=>[4,1,2,5,3]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,+,+,-,+]=>[1,2,3,5,4]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [+,+,+,5,4]=>[1,2,3,5,4]=>[5,1,2,3,4]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [+,+,4,3,+]=>[1,2,4,5,3]=>[4,1,2,5,3]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,+,5,3,4]=>[1,2,4,5,3]=>[4,1,2,5,3]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,+,5,+,3]=>[1,2,4,5,3]=>[4,1,2,5,3]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,3,2,+,+]=>[1,3,4,5,2]=>[3,1,4,5,2]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,4,2,3,+]=>[1,3,4,5,2]=>[3,1,4,5,2]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,4,+,2,+]=>[1,3,4,5,2]=>[3,1,4,5,2]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,5,2,3,4]=>[1,3,4,5,2]=>[3,1,4,5,2]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,5,2,+,3]=>[1,3,4,5,2]=>[3,1,4,5,2]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,5,+,2,4]=>[1,3,4,5,2]=>[3,1,4,5,2]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [+,5,+,+,2]=>[1,3,4,5,2]=>[3,1,4,5,2]=>([(0,4),(1,4),(2,3),(3,4)],5)=>2 [2,1,+,+,+]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [3,1,2,+,+]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [3,+,1,+,+]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [4,1,2,3,+]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [4,1,+,2,+]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [4,+,1,3,+]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [4,+,+,1,+]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [5,1,2,3,4]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [5,1,2,+,3]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [5,1,+,2,4]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [5,1,+,+,2]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [5,+,1,3,4]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [5,+,1,+,3]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [5,+,+,1,4]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [5,+,+,+,1]=>[2,3,4,5,1]=>[2,3,4,5,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>1 [-,+,+,+,+,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [+,-,+,+,+,+]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,+,-,+,+,+]=>[1,2,4,5,6,3]=>[4,1,2,5,6,3]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2 [+,+,+,-,+,+]=>[1,2,3,5,6,4]=>[5,1,2,3,6,4]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,+,+,+,-,+]=>[1,2,3,4,6,5]=>[6,1,2,3,4,5]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [+,+,+,+,6,5]=>[1,2,3,4,6,5]=>[6,1,2,3,4,5]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [+,+,+,5,4,+]=>[1,2,3,5,6,4]=>[5,1,2,3,6,4]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,+,+,6,4,5]=>[1,2,3,5,6,4]=>[5,1,2,3,6,4]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,+,+,6,+,4]=>[1,2,3,5,6,4]=>[5,1,2,3,6,4]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,+,4,3,+,+]=>[1,2,4,5,6,3]=>[4,1,2,5,6,3]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2 [+,+,5,3,4,+]=>[1,2,4,5,6,3]=>[4,1,2,5,6,3]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2 [+,+,5,+,3,+]=>[1,2,4,5,6,3]=>[4,1,2,5,6,3]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2 [+,+,6,3,4,5]=>[1,2,4,5,6,3]=>[4,1,2,5,6,3]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2 [+,+,6,3,+,4]=>[1,2,4,5,6,3]=>[4,1,2,5,6,3]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2 [+,+,6,+,3,5]=>[1,2,4,5,6,3]=>[4,1,2,5,6,3]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2 [+,+,6,+,+,3]=>[1,2,4,5,6,3]=>[4,1,2,5,6,3]=>([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2 [+,3,2,+,+,+]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,4,2,3,+,+]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,4,+,2,+,+]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,5,2,3,4,+]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,5,2,+,3,+]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,5,+,2,4,+]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,5,+,+,2,+]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,6,2,3,4,5]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,6,2,3,+,4]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,6,2,+,3,5]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,6,2,+,+,3]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,6,+,2,4,5]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,6,+,2,+,4]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,6,+,+,2,5]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [+,6,+,+,+,2]=>[1,3,4,5,6,2]=>[3,1,4,5,6,2]=>([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2 [2,1,+,+,+,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [3,1,2,+,+,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [3,+,1,+,+,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [4,1,2,3,+,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [4,1,+,2,+,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [4,+,1,3,+,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [4,+,+,1,+,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [5,1,2,3,4,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [5,1,2,+,3,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [5,1,+,2,4,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [5,1,+,+,2,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [5,+,1,3,4,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [5,+,1,+,3,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [5,+,+,1,4,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [5,+,+,+,1,+]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,1,2,3,4,5]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,1,2,3,+,4]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,1,2,+,3,5]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,1,2,+,+,3]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,1,+,2,4,5]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,1,+,2,+,4]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,1,+,+,2,5]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,1,+,+,+,2]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,+,1,3,4,5]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,+,1,3,+,4]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,+,1,+,3,5]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,+,1,+,+,3]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,+,+,1,4,5]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,+,+,1,+,4]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,+,+,+,1,5]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1 [6,+,+,+,+,1]=>[2,3,4,5,6,1]=>[2,3,4,5,6,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>1
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 radius of a connected graph.
This is the minimum eccentricity of any vertex.
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
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.
  • 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 $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
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
upper permutation
Description
The upper bound in the Grassmann interval corresponding to the decorated permutation.
Let $I$ be the anti-exceedance set of a decorated permutation $w$. Let $v$ be the $k$-Grassmannian permutation determined by $v[k] = w^{-1}(I)$ and let $u$ be the permutation satisfying $u = wv$. Then $[u, v]$ is the Grassmann interval corresponding to $w$.
This map returns $v$.