Identifier
Values
[+] => [1] => [1,0] => [[1],[]] => 1
[-] => [1] => [1,0] => [[1],[]] => 1
[+,+] => [1,2] => [1,0,1,0] => [[1,1],[]] => 1
[-,+] => [1,2] => [1,0,1,0] => [[1,1],[]] => 1
[+,-] => [1,2] => [1,0,1,0] => [[1,1],[]] => 1
[-,-] => [1,2] => [1,0,1,0] => [[1,1],[]] => 1
[2,1] => [2,1] => [1,1,0,0] => [[2],[]] => 1
[+,+,+] => [1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[-,+,+] => [1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[+,-,+] => [1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[+,+,-] => [1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[-,-,+] => [1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[-,+,-] => [1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[+,-,-] => [1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[-,-,-] => [1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => 1
[+,3,2] => [1,3,2] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[-,3,2] => [1,3,2] => [1,0,1,1,0,0] => [[2,1],[]] => 1
[2,1,+] => [2,1,3] => [1,1,0,0,1,0] => [[2,2],[1]] => 1
[2,1,-] => [2,1,3] => [1,1,0,0,1,0] => [[2,2],[1]] => 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0] => [[3],[]] => 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0] => [[2,2],[]] => 1
[3,+,1] => [3,2,1] => [1,1,1,0,0,0] => [[2,2],[]] => 1
[3,-,1] => [3,2,1] => [1,1,1,0,0,0] => [[2,2],[]] => 1
[+,+,+,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[-,+,+,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[+,-,+,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[+,+,-,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[+,+,+,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[-,-,+,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[-,+,-,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[-,+,+,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[+,-,-,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[+,-,+,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[+,+,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[-,-,-,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[-,-,+,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[-,+,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[+,-,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[-,-,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 1
[+,+,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[-,+,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[+,-,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[-,-,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 1
[+,3,2,+] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 1
[-,3,2,+] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 1
[+,3,2,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 1
[-,3,2,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 1
[+,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[-,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 1
[+,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[-,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[+,4,+,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[-,4,+,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[+,4,-,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[-,4,-,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 1
[2,1,+,+] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 1
[2,1,-,+] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 1
[2,1,+,-] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 1
[2,1,-,-] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 1
[2,3,1,+] => [2,3,1,4] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 1
[2,3,1,-] => [2,3,1,4] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 1
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => [[4],[]] => 1
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 1
[2,4,+,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 1
[2,4,-,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 1
[3,1,2,+] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[3,1,2,-] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 1
[3,+,1,+] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[3,-,1,+] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[3,+,1,-] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[3,-,1,-] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[3,+,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 1
[3,-,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 1
[+,+,+,+,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,+,+,+,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,-,+,+,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,+,-,+,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,+,+,-,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,+,+,+,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,-,+,+,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,+,-,+,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,+,+,-,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,+,+,+,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,-,-,+,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,-,+,-,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,-,+,+,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,+,-,-,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,+,-,+,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,+,+,-,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,-,-,+,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,-,+,-,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,-,+,+,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,+,-,-,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,+,-,+,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,+,+,-,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,-,-,-,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,-,-,+,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,-,+,-,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,+,-,-,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
>>> Load all 164 entries. <<<
[-,-,-,-,+] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,-,-,+,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,-,+,-,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,+,-,-,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,-,-,-,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[-,-,-,-,-] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 1
[+,+,+,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[-,+,+,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[+,-,+,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[+,+,-,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[-,-,+,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[-,+,-,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[+,-,-,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[-,-,-,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 1
[+,+,4,3,+] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[-,+,4,3,+] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[+,-,4,3,+] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[+,+,4,3,-] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[-,-,4,3,+] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[-,+,4,3,-] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[+,-,4,3,-] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[-,-,4,3,-] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[+,+,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 1
[-,+,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 1
[+,-,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 1
[-,-,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 1
[+,3,2,+,+] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[-,3,2,+,+] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[+,3,2,-,+] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[+,3,2,+,-] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[-,3,2,-,+] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[-,3,2,+,-] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[+,3,2,-,-] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[-,3,2,-,-] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 1
[+,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => 1
[-,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => 1
[+,3,4,2,+] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 1
[-,3,4,2,+] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 1
[+,3,4,2,-] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 1
[-,3,4,2,-] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 1
[+,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => 1
[-,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => 1
[2,1,+,+,+] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[2,1,-,+,+] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[2,1,+,-,+] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[2,1,+,+,-] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[2,1,-,-,+] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[2,1,-,+,-] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[2,1,+,-,-] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[2,1,-,-,-] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 1
[2,1,+,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 1
[2,1,-,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 1
[2,1,4,3,+] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => 1
[2,1,4,3,-] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => 1
[2,1,4,5,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => 1
[2,3,1,+,+] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 1
[2,3,1,-,+] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 1
[2,3,1,+,-] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 1
[2,3,1,-,-] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 1
[2,3,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => 1
[2,3,4,1,+] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 1
[2,3,4,1,-] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 1
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
Description
The number of connected components of a skew partition.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.
Map
permutation
Description
The underlying permutation of the decorated permutation.