Identifier
Values
0 => [2] => [1,1,0,0] => 0
1 => [1,1] => [1,0,1,0] => 1
00 => [3] => [1,1,1,0,0,0] => 0
01 => [2,1] => [1,1,0,0,1,0] => 2
10 => [1,2] => [1,0,1,1,0,0] => 1
11 => [1,1,1] => [1,0,1,0,1,0] => 3
000 => [4] => [1,1,1,1,0,0,0,0] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => 2
010 => [2,2] => [1,1,0,0,1,1,0,0] => 3
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 5
100 => [1,3] => [1,0,1,1,1,0,0,0] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 6
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 4
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 5
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 7
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 6
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 5
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 6
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 5
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 5
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 9
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 3
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 9
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 12
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 6
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 11
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 10
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 14
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 4
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 6
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 8
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 10
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 9
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 7
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 8
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 6
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 10
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 15
=> [1] => [1,0] => 0
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Description
The dinv of a Dyck path.
Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see St000012The area of a Dyck path.).
The dinv statistic of $D$ is
$$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$
Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose arm length is one larger or equal to its leg length.
There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2].
Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by
$$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.