Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000796: Permutations ⟶ ℤ
Values
00 => [2] => [1,1,0,0] => [1,2] => 0
01 => [1,1] => [1,0,1,0] => [2,1] => 1
10 => [1,1] => [1,0,1,0] => [2,1] => 1
11 => [2] => [1,1,0,0] => [1,2] => 0
000 => [3] => [1,1,1,0,0,0] => [1,2,3] => 0
001 => [2,1] => [1,1,0,0,1,0] => [1,3,2] => 2
010 => [1,1,1] => [1,0,1,0,1,0] => [2,3,1] => 1
011 => [1,2] => [1,0,1,1,0,0] => [2,1,3] => 1
100 => [1,2] => [1,0,1,1,0,0] => [2,1,3] => 1
101 => [1,1,1] => [1,0,1,0,1,0] => [2,3,1] => 1
110 => [2,1] => [1,1,0,0,1,0] => [1,3,2] => 2
111 => [3] => [1,1,1,0,0,0] => [1,2,3] => 0
0000 => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 2
0011 => [2,2] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => 3
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => 2
1111 => [4] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 2
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 2
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 2
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 4
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 2
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 3
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 3
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 3
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 3
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => 3
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => 3
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => 3
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => 1
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => 3
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => 1
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => 2
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => 4
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => 2
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => 2
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => 2
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => 2
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => 2
11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 0
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 0
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => 2
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => 2
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => 2
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,2,4,5,3,6] => 2
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => 2
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => 4
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => 2
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,4,2,5,6] => 2
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => 4
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => 2
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,4,5,2,6] => 2
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => 4
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => 4
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => 4
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => 2
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => 1
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => 3
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => 3
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => 3
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => 1
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => 1
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => 3
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => 1
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => 3
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => 5
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => 3
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => 3
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => 3
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => 3
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => 3
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => 1
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => 1
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => 3
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => 3
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => 3
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => 3
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => 3
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => 5
100111 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => 3
101000 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => 1
>>> Load all 188 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 stat' of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns $(\underline{13}2)$, $(\underline{31}2)$, $(\underline{32}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
According to [1], this is the sum of the number of occurrences of the vincular patterns $(\underline{13}2)$, $(\underline{31}2)$, $(\underline{32}2)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
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!