Identifier
Values
=>
Cc0014;cc-rep
[1]=>[1]=>[1]=>([],1)=>0 [1,2]=>[1,2]=>[1,2]=>([(0,1)],2)=>0 [2,1]=>[2,1]=>[2,1]=>([],2)=>0 [2,1,3]=>[2,1,3]=>[2,1,3]=>([(0,2),(1,2)],3)=>0 [3,1,2]=>[3,1,2]=>[1,3,2]=>([(0,1),(0,2)],3)=>0 [3,2,1]=>[3,2,1]=>[3,2,1]=>([],3)=>0 [2,1,3,4]=>[2,1,4,3]=>[4,2,1,3]=>([(1,3),(2,3)],4)=>0 [2,1,4,3]=>[2,1,4,3]=>[4,2,1,3]=>([(1,3),(2,3)],4)=>0 [2,3,1,4]=>[2,4,1,3]=>[2,1,4,3]=>([(0,2),(0,3),(1,2),(1,3)],4)=>2 [2,4,1,3]=>[2,4,1,3]=>[2,1,4,3]=>([(0,2),(0,3),(1,2),(1,3)],4)=>2 [3,2,1,4]=>[3,2,1,4]=>[3,2,1,4]=>([(0,3),(1,3),(2,3)],4)=>1 [3,2,4,1]=>[3,2,4,1]=>[3,2,4,1]=>([(1,3),(2,3)],4)=>0 [3,4,1,2]=>[3,4,1,2]=>[1,3,4,2]=>([(0,2),(0,3),(3,1)],4)=>0 [4,2,1,3]=>[4,2,1,3]=>[2,4,1,3]=>([(0,3),(1,2),(1,3)],4)=>0 [4,3,1,2]=>[4,3,1,2]=>[1,4,3,2]=>([(0,1),(0,2),(0,3)],4)=>1 [4,3,2,1]=>[4,3,2,1]=>[4,3,2,1]=>([],4)=>0 [2,1,3,4,5]=>[2,1,5,4,3]=>[5,4,2,1,3]=>([(2,4),(3,4)],5)=>0 [2,1,3,5,4]=>[2,1,5,4,3]=>[5,4,2,1,3]=>([(2,4),(3,4)],5)=>0 [2,1,4,3,5]=>[2,1,5,4,3]=>[5,4,2,1,3]=>([(2,4),(3,4)],5)=>0 [2,1,4,5,3]=>[2,1,5,4,3]=>[5,4,2,1,3]=>([(2,4),(3,4)],5)=>0 [2,1,5,3,4]=>[2,1,5,4,3]=>[5,4,2,1,3]=>([(2,4),(3,4)],5)=>0 [2,1,5,4,3]=>[2,1,5,4,3]=>[5,4,2,1,3]=>([(2,4),(3,4)],5)=>0 [2,3,4,1,5]=>[2,5,4,1,3]=>[2,5,1,4,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4)],5)=>2 [2,3,5,1,4]=>[2,5,4,1,3]=>[2,5,1,4,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4)],5)=>2 [2,4,3,1,5]=>[2,5,4,1,3]=>[2,5,1,4,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4)],5)=>2 [2,4,5,1,3]=>[2,5,4,1,3]=>[2,5,1,4,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4)],5)=>2 [2,5,3,1,4]=>[2,5,4,1,3]=>[2,5,1,4,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4)],5)=>2 [2,5,4,1,3]=>[2,5,4,1,3]=>[2,5,1,4,3]=>([(0,3),(0,4),(1,2),(1,3),(1,4)],5)=>2 [3,2,4,1,5]=>[3,2,5,1,4]=>[3,2,1,5,4]=>([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)=>3 [3,2,4,5,1]=>[3,2,5,4,1]=>[5,3,2,4,1]=>([(2,4),(3,4)],5)=>0 [3,2,5,1,4]=>[3,2,5,1,4]=>[3,2,1,5,4]=>([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)=>3 [3,2,5,4,1]=>[3,2,5,4,1]=>[5,3,2,4,1]=>([(2,4),(3,4)],5)=>0 [3,4,1,2,5]=>[3,5,1,4,2]=>[3,5,1,4,2]=>([(0,3),(0,4),(1,2),(1,4)],5)=>0 [3,4,1,5,2]=>[3,5,1,4,2]=>[3,5,1,4,2]=>([(0,3),(0,4),(1,2),(1,4)],5)=>0 [3,4,2,1,5]=>[3,5,2,1,4]=>[3,2,5,1,4]=>([(0,4),(1,3),(1,4),(2,3),(2,4)],5)=>2 [3,4,5,1,2]=>[3,5,4,1,2]=>[1,5,3,4,2]=>([(0,2),(0,3),(0,4),(4,1)],5)=>1 [3,5,1,2,4]=>[3,5,1,4,2]=>[3,5,1,4,2]=>([(0,3),(0,4),(1,2),(1,4)],5)=>0 [3,5,1,4,2]=>[3,5,1,4,2]=>[3,5,1,4,2]=>([(0,3),(0,4),(1,2),(1,4)],5)=>0 [3,5,2,1,4]=>[3,5,2,1,4]=>[3,2,5,1,4]=>([(0,4),(1,3),(1,4),(2,3),(2,4)],5)=>2 [3,5,4,1,2]=>[3,5,4,1,2]=>[1,5,3,4,2]=>([(0,2),(0,3),(0,4),(4,1)],5)=>1 [4,2,1,3,5]=>[4,2,1,5,3]=>[4,5,2,1,3]=>([(0,4),(1,4),(2,3)],5)=>0 [4,2,1,5,3]=>[4,2,1,5,3]=>[4,5,2,1,3]=>([(0,4),(1,4),(2,3)],5)=>0 [4,2,3,1,5]=>[4,2,5,1,3]=>[2,4,1,5,3]=>([(0,3),(0,4),(1,2),(1,3),(2,4)],5)=>2 [4,2,5,1,3]=>[4,2,5,1,3]=>[2,4,1,5,3]=>([(0,3),(0,4),(1,2),(1,3),(2,4)],5)=>2 [4,3,1,2,5]=>[4,3,1,5,2]=>[4,3,5,1,2]=>([(0,4),(1,4),(2,3)],5)=>0 [4,3,1,5,2]=>[4,3,1,5,2]=>[4,3,5,1,2]=>([(0,4),(1,4),(2,3)],5)=>0 [4,3,2,1,5]=>[4,3,2,1,5]=>[4,3,2,1,5]=>([(0,4),(1,4),(2,4),(3,4)],5)=>2 [4,3,5,1,2]=>[4,3,5,1,2]=>[1,4,3,5,2]=>([(0,1),(0,2),(0,3),(2,4),(3,4)],5)=>1 [4,3,5,2,1]=>[4,3,5,2,1]=>[4,3,5,2,1]=>([(2,4),(3,4)],5)=>0 [4,5,1,2,3]=>[4,5,1,3,2]=>[4,1,5,3,2]=>([(0,4),(1,2),(1,3),(1,4)],5)=>1 [4,5,1,3,2]=>[4,5,1,3,2]=>[4,1,5,3,2]=>([(0,4),(1,2),(1,3),(1,4)],5)=>1 [4,5,2,1,3]=>[4,5,2,1,3]=>[2,4,5,1,3]=>([(0,4),(1,2),(1,4),(2,3)],5)=>0 [4,5,3,1,2]=>[4,5,3,1,2]=>[1,4,5,3,2]=>([(0,2),(0,3),(0,4),(4,1)],5)=>1 [5,2,3,1,4]=>[5,2,4,1,3]=>[2,1,5,4,3]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)=>3 [5,2,4,1,3]=>[5,2,4,1,3]=>[2,1,5,4,3]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)=>3 [5,3,2,1,4]=>[5,3,2,1,4]=>[3,5,2,1,4]=>([(0,4),(1,4),(2,3),(2,4)],5)=>1 [5,3,4,1,2]=>[5,3,4,1,2]=>[1,3,5,4,2]=>([(0,3),(0,4),(4,1),(4,2)],5)=>1 [5,4,2,1,3]=>[5,4,2,1,3]=>[2,5,4,1,3]=>([(0,4),(1,2),(1,3),(1,4)],5)=>1 [5,4,3,1,2]=>[5,4,3,1,2]=>[1,5,4,3,2]=>([(0,1),(0,2),(0,3),(0,4)],5)=>2 [5,4,3,2,1]=>[5,4,3,2,1]=>[5,4,3,2,1]=>([],5)=>0 [2,1,3,4,5,6]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,3,4,6,5]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,3,5,4,6]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,3,5,6,4]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,3,6,4,5]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,3,6,5,4]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,4,3,5,6]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,4,3,6,5]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,4,5,3,6]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,4,5,6,3]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,4,6,3,5]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,4,6,5,3]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,5,3,4,6]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,5,3,6,4]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,5,4,3,6]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,5,4,6,3]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,5,6,3,4]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,5,6,4,3]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,6,3,4,5]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,6,3,5,4]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,6,4,3,5]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,6,4,5,3]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,6,5,3,4]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [2,1,6,5,4,3]=>[2,1,6,5,4,3]=>[6,5,4,2,1,3]=>([(3,5),(4,5)],6)=>0 [3,2,4,5,6,1]=>[3,2,6,5,4,1]=>[6,5,3,2,4,1]=>([(3,5),(4,5)],6)=>0 [3,2,4,6,5,1]=>[3,2,6,5,4,1]=>[6,5,3,2,4,1]=>([(3,5),(4,5)],6)=>0 [3,2,5,4,6,1]=>[3,2,6,5,4,1]=>[6,5,3,2,4,1]=>([(3,5),(4,5)],6)=>0 [3,2,5,6,4,1]=>[3,2,6,5,4,1]=>[6,5,3,2,4,1]=>([(3,5),(4,5)],6)=>0 [3,2,6,4,5,1]=>[3,2,6,5,4,1]=>[6,5,3,2,4,1]=>([(3,5),(4,5)],6)=>0 [3,2,6,5,4,1]=>[3,2,6,5,4,1]=>[6,5,3,2,4,1]=>([(3,5),(4,5)],6)=>0 [4,2,1,3,5,6]=>[4,2,1,6,5,3]=>[6,4,5,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [4,2,1,3,6,5]=>[4,2,1,6,5,3]=>[6,4,5,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [4,2,1,5,3,6]=>[4,2,1,6,5,3]=>[6,4,5,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [4,2,1,5,6,3]=>[4,2,1,6,5,3]=>[6,4,5,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [4,2,1,6,3,5]=>[4,2,1,6,5,3]=>[6,4,5,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [4,2,1,6,5,3]=>[4,2,1,6,5,3]=>[6,4,5,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [4,3,1,2,5,6]=>[4,3,1,6,5,2]=>[6,4,3,5,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [4,3,1,2,6,5]=>[4,3,1,6,5,2]=>[6,4,3,5,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [4,3,1,5,2,6]=>[4,3,1,6,5,2]=>[6,4,3,5,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [4,3,1,5,6,2]=>[4,3,1,6,5,2]=>[6,4,3,5,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [4,3,1,6,2,5]=>[4,3,1,6,5,2]=>[6,4,3,5,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [4,3,1,6,5,2]=>[4,3,1,6,5,2]=>[6,4,3,5,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [4,3,5,6,2,1]=>[4,3,6,5,2,1]=>[6,4,3,5,2,1]=>([(3,5),(4,5)],6)=>0 [4,3,6,5,2,1]=>[4,3,6,5,2,1]=>[6,4,3,5,2,1]=>([(3,5),(4,5)],6)=>0 [5,2,1,3,4,6]=>[5,2,1,6,4,3]=>[5,6,4,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [5,2,1,3,6,4]=>[5,2,1,6,4,3]=>[5,6,4,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [5,2,1,4,3,6]=>[5,2,1,6,4,3]=>[5,6,4,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [5,2,1,4,6,3]=>[5,2,1,6,4,3]=>[5,6,4,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [5,2,1,6,3,4]=>[5,2,1,6,4,3]=>[5,6,4,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [5,2,1,6,4,3]=>[5,2,1,6,4,3]=>[5,6,4,2,1,3]=>([(1,5),(2,5),(3,4)],6)=>0 [5,3,2,4,6,1]=>[5,3,2,6,4,1]=>[5,6,3,2,4,1]=>([(1,5),(2,5),(3,4)],6)=>0 [5,3,2,6,4,1]=>[5,3,2,6,4,1]=>[5,6,3,2,4,1]=>([(1,5),(2,5),(3,4)],6)=>0 [5,4,1,2,3,6]=>[5,4,1,6,3,2]=>[5,4,6,3,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [5,4,1,2,6,3]=>[5,4,1,6,3,2]=>[5,4,6,3,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [5,4,1,3,2,6]=>[5,4,1,6,3,2]=>[5,4,6,3,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [5,4,1,3,6,2]=>[5,4,1,6,3,2]=>[5,4,6,3,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [5,4,1,6,2,3]=>[5,4,1,6,3,2]=>[5,4,6,3,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [5,4,1,6,3,2]=>[5,4,1,6,3,2]=>[5,4,6,3,1,2]=>([(1,5),(2,5),(3,4)],6)=>0 [5,4,2,1,3,6]=>[5,4,2,1,6,3]=>[5,4,6,2,1,3]=>([(0,5),(1,5),(2,4),(3,4)],6)=>0 [5,4,2,1,6,3]=>[5,4,2,1,6,3]=>[5,4,6,2,1,3]=>([(0,5),(1,5),(2,4),(3,4)],6)=>0 [5,4,2,3,6,1]=>[5,4,2,6,3,1]=>[5,4,6,2,3,1]=>([(1,5),(2,5),(3,4)],6)=>0 [5,4,2,6,3,1]=>[5,4,2,6,3,1]=>[5,4,6,2,3,1]=>([(1,5),(2,5),(3,4)],6)=>0 [5,4,3,2,1,6]=>[5,4,3,2,1,6]=>[5,4,3,2,1,6]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>3 [5,4,6,3,2,1]=>[5,4,6,3,2,1]=>[5,4,6,3,2,1]=>([(3,5),(4,5)],6)=>0 [6,5,4,3,2,1]=>[6,5,4,3,2,1]=>[6,5,4,3,2,1]=>([],6)=>0
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 interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
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.).