Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤ
Values
0 => [1] => [1,0] => [] => 0
1 => [1] => [1,0] => [] => 0
00 => [2] => [1,1,0,0] => [] => 0
01 => [1,1] => [1,0,1,0] => [1] => 1
10 => [1,1] => [1,0,1,0] => [1] => 1
11 => [2] => [1,1,0,0] => [] => 0
000 => [3] => [1,1,1,0,0,0] => [] => 0
001 => [2,1] => [1,1,0,0,1,0] => [2] => 1
010 => [1,1,1] => [1,0,1,0,1,0] => [2,1] => 2
011 => [1,2] => [1,0,1,1,0,0] => [1,1] => 2
100 => [1,2] => [1,0,1,1,0,0] => [1,1] => 2
101 => [1,1,1] => [1,0,1,0,1,0] => [2,1] => 2
110 => [2,1] => [1,1,0,0,1,0] => [2] => 1
111 => [3] => [1,1,1,0,0,0] => [] => 0
0000 => [4] => [1,1,1,1,0,0,0,0] => [] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0] => [3] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => [3,2] => 2
0011 => [2,2] => [1,1,0,0,1,1,0,0] => [2,2] => 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,2,1] => 3
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [3,2,1] => 3
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,1] => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1] => 3
1000 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1] => 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,1] => 3
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [3,2,1] => 3
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,2,1] => 3
1100 => [2,2] => [1,1,0,0,1,1,0,0] => [2,2] => 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => [3,2] => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0] => [3] => 1
1111 => [4] => [1,1,1,1,0,0,0,0] => [] => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [] => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4] => 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 2
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 2
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 3
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 3
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 3
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 3
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 4
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 4
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 4
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 4
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 4
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 4
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 4
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 4
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 4
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 4
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 4
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 4
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 4
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 4
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 4
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 4
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 3
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 3
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 3
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 3
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 2
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 2
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4] => 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => [] => 0
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [] => 0
000001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => 1
000010 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => 2
000011 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 2
000100 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => 3
000101 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [5,4,3] => 3
000110 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 3
000111 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 3
001000 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 4
001001 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => 4
001010 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [5,4,3,2] => 4
001011 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => 4
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 4
001101 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => 4
001110 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 4
001111 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 4
010000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 5
010001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 5
010010 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [5,4,2,2,1] => 5
010011 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 5
010100 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [4,4,3,2,1] => 5
010101 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 5
010110 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,3,3,2,1] => 5
010111 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 5
011000 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 5
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 5
011010 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 5
011011 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 5
011100 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 5
011101 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 5
011110 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 5
011111 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 5
100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 5
100001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 5
100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 5
100011 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 5
100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 5
100101 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,1] => 5
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 5
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Description
The length of the partition.
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 $w$.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
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