Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001153: Set partitions ⟶ ℤ
Values
0 => [1] => [1,0] => {{1}} => 0
1 => [1] => [1,0] => {{1}} => 0
00 => [2] => [1,1,0,0] => {{1,2}} => 0
01 => [1,1] => [1,0,1,0] => {{1},{2}} => 1
10 => [1,1] => [1,0,1,0] => {{1},{2}} => 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] => {{1,2},{3}} => 0
010 => [1,1,1] => [1,0,1,0,1,0] => {{1},{2},{3}} => 1
011 => [1,2] => [1,0,1,1,0,0] => {{1},{2,3}} => 1
100 => [1,2] => [1,0,1,1,0,0] => {{1},{2,3}} => 1
101 => [1,1,1] => [1,0,1,0,1,0] => {{1},{2},{3}} => 1
110 => [2,1] => [1,1,0,0,1,0] => {{1,2},{3}} => 0
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] => {{1,2,3},{4}} => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => {{1,2},{3},{4}} => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0] => {{1,2},{3,4}} => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => {{1},{2},{3,4}} => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => {{1},{2},{3},{4}} => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => {{1},{2,3},{4}} => 2
0111 => [1,3] => [1,0,1,1,1,0,0,0] => {{1},{2,3,4}} => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0] => {{1},{2,3,4}} => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => {{1},{2,3},{4}} => 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => {{1},{2},{3},{4}} => 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => {{1},{2},{3,4}} => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0] => {{1,2},{3,4}} => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => {{1,2},{3},{4}} => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0] => {{1,2,3},{4}} => 1
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] => {{1,2,3,4},{5}} => 0
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => {{1,2,3},{4},{5}} => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => {{1,2,3},{4,5}} => 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => {{1,2},{3},{4,5}} => 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => {{1,2},{3},{4},{5}} => 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => {{1,2},{3,4},{5}} => 0
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => {{1,2},{3,4,5}} => 0
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3,4,5}} => 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3,4},{5}} => 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5}} => 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4,5}} => 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => {{1},{2,3},{4,5}} => 2
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => {{1},{2,3},{4},{5}} => 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => {{1},{2,3,4},{5}} => 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => {{1},{2,3,4,5}} => 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => {{1},{2,3,4,5}} => 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => {{1},{2,3,4},{5}} => 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => {{1},{2,3},{4},{5}} => 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => {{1},{2,3},{4,5}} => 2
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4,5}} => 2
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5}} => 2
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3,4},{5}} => 1
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3,4,5}} => 1
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => {{1,2},{3,4,5}} => 0
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => {{1,2},{3,4},{5}} => 0
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => {{1,2},{3},{4},{5}} => 1
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => {{1,2},{3},{4,5}} => 1
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => {{1,2,3},{4,5}} => 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => {{1,2,3},{4},{5}} => 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => {{1,2,3,4},{5}} => 0
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] => {{1,2,3,4,5},{6}} => 1
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => {{1,2,3,4},{5},{6}} => 1
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => {{1,2,3,4},{5,6}} => 0
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => {{1,2,3},{4},{5,6}} => 1
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => {{1,2,3},{4},{5},{6}} => 2
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => {{1,2,3},{4,5},{6}} => 2
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => {{1,2,3},{4,5,6}} => 1
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => {{1,2},{3},{4,5,6}} => 1
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => {{1,2},{3},{4,5},{6}} => 2
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => {{1,2},{3},{4},{5},{6}} => 2
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => {{1,2},{3},{4},{5,6}} => 1
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => {{1,2},{3,4},{5,6}} => 0
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => {{1,2},{3,4},{5},{6}} => 1
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => {{1,2},{3,4,5},{6}} => 1
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => {{1,2},{3,4,5,6}} => 0
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => {{1},{2},{3,4,5,6}} => 1
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => {{1},{2},{3,4,5},{6}} => 2
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => {{1},{2},{3,4},{5},{6}} => 2
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => {{1},{2},{3,4},{5,6}} => 1
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4},{5,6}} => 2
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5},{6}} => 3
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3},{4,5},{6}} => 3
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3},{4,5,6}} => 2
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => {{1},{2,3},{4,5,6}} => 2
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => {{1},{2,3},{4,5},{6}} => 3
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => {{1},{2,3},{4},{5},{6}} => 3
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => {{1},{2,3},{4},{5,6}} => 2
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => {{1},{2,3,4},{5,6}} => 1
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => {{1},{2,3,4},{5},{6}} => 2
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => {{1},{2,3,4,5},{6}} => 2
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => {{1},{2,3,4,5,6}} => 1
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => {{1},{2,3,4,5,6}} => 1
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => {{1},{2,3,4,5},{6}} => 2
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => {{1},{2,3,4},{5},{6}} => 2
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => {{1},{2,3,4},{5,6}} => 1
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => {{1},{2,3},{4},{5,6}} => 2
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => {{1},{2,3},{4},{5},{6}} => 3
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => {{1},{2,3},{4,5},{6}} => 3
>>> Load all 276 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 number of blocks with even minimum in a set partition.
Map
to noncrossing partition
Description
Biane's map to noncrossing set partitions.
Map
bounce path
Description
The bounce path determined by an integer composition.
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!