- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001486: Integer compositions ⟶ ℤ

Values

[1] => [1] => 1 => [1,1] => 2

[1,2] => [2] => 10 => [1,2] => 3

[2,1] => [1,1] => 11 => [1,1,1] => 2

[1,2,3] => [3] => 100 => [1,3] => 3

[1,3,2] => [2,1] => 101 => [1,2,1] => 4

[2,1,3] => [1,2] => 110 => [1,1,2] => 3

[2,3,1] => [2,1] => 101 => [1,2,1] => 4

[3,1,2] => [1,2] => 110 => [1,1,2] => 3

[3,2,1] => [1,1,1] => 111 => [1,1,1,1] => 2

[1,2,3,4] => [4] => 1000 => [1,4] => 3

[1,2,4,3] => [3,1] => 1001 => [1,3,1] => 4

[1,3,2,4] => [2,2] => 1010 => [1,2,2] => 5

[1,3,4,2] => [3,1] => 1001 => [1,3,1] => 4

[1,4,2,3] => [2,2] => 1010 => [1,2,2] => 5

[1,4,3,2] => [2,1,1] => 1011 => [1,2,1,1] => 4

[2,1,3,4] => [1,3] => 1100 => [1,1,3] => 3

[2,1,4,3] => [1,2,1] => 1101 => [1,1,2,1] => 4

[2,3,1,4] => [2,2] => 1010 => [1,2,2] => 5

[2,3,4,1] => [3,1] => 1001 => [1,3,1] => 4

[2,4,1,3] => [2,2] => 1010 => [1,2,2] => 5

[2,4,3,1] => [2,1,1] => 1011 => [1,2,1,1] => 4

[3,1,2,4] => [1,3] => 1100 => [1,1,3] => 3

[3,1,4,2] => [1,2,1] => 1101 => [1,1,2,1] => 4

[3,2,1,4] => [1,1,2] => 1110 => [1,1,1,2] => 3

[3,2,4,1] => [1,2,1] => 1101 => [1,1,2,1] => 4

[3,4,1,2] => [2,2] => 1010 => [1,2,2] => 5

[3,4,2,1] => [2,1,1] => 1011 => [1,2,1,1] => 4

[4,1,2,3] => [1,3] => 1100 => [1,1,3] => 3

[4,1,3,2] => [1,2,1] => 1101 => [1,1,2,1] => 4

[4,2,1,3] => [1,1,2] => 1110 => [1,1,1,2] => 3

[4,2,3,1] => [1,2,1] => 1101 => [1,1,2,1] => 4

[4,3,1,2] => [1,1,2] => 1110 => [1,1,1,2] => 3

[4,3,2,1] => [1,1,1,1] => 1111 => [1,1,1,1,1] => 2

[1,2,3,4,5] => [5] => 10000 => [1,5] => 3

[1,2,3,5,4] => [4,1] => 10001 => [1,4,1] => 4

[1,2,4,3,5] => [3,2] => 10010 => [1,3,2] => 5

[1,2,4,5,3] => [4,1] => 10001 => [1,4,1] => 4

[1,2,5,3,4] => [3,2] => 10010 => [1,3,2] => 5

[1,2,5,4,3] => [3,1,1] => 10011 => [1,3,1,1] => 4

[1,3,2,4,5] => [2,3] => 10100 => [1,2,3] => 5

[1,3,2,5,4] => [2,2,1] => 10101 => [1,2,2,1] => 6

[1,3,4,2,5] => [3,2] => 10010 => [1,3,2] => 5

[1,3,4,5,2] => [4,1] => 10001 => [1,4,1] => 4

[1,3,5,2,4] => [3,2] => 10010 => [1,3,2] => 5

[1,3,5,4,2] => [3,1,1] => 10011 => [1,3,1,1] => 4

[1,4,2,3,5] => [2,3] => 10100 => [1,2,3] => 5

[1,4,2,5,3] => [2,2,1] => 10101 => [1,2,2,1] => 6

[1,4,3,2,5] => [2,1,2] => 10110 => [1,2,1,2] => 5

[1,4,3,5,2] => [2,2,1] => 10101 => [1,2,2,1] => 6

[1,4,5,2,3] => [3,2] => 10010 => [1,3,2] => 5

[1,4,5,3,2] => [3,1,1] => 10011 => [1,3,1,1] => 4

[1,5,2,3,4] => [2,3] => 10100 => [1,2,3] => 5

[1,5,2,4,3] => [2,2,1] => 10101 => [1,2,2,1] => 6

[1,5,3,2,4] => [2,1,2] => 10110 => [1,2,1,2] => 5

[1,5,3,4,2] => [2,2,1] => 10101 => [1,2,2,1] => 6

[1,5,4,2,3] => [2,1,2] => 10110 => [1,2,1,2] => 5

[1,5,4,3,2] => [2,1,1,1] => 10111 => [1,2,1,1,1] => 4

[2,1,3,4,5] => [1,4] => 11000 => [1,1,4] => 3

[2,1,3,5,4] => [1,3,1] => 11001 => [1,1,3,1] => 4

[2,1,4,3,5] => [1,2,2] => 11010 => [1,1,2,2] => 5

[2,1,4,5,3] => [1,3,1] => 11001 => [1,1,3,1] => 4

[2,1,5,3,4] => [1,2,2] => 11010 => [1,1,2,2] => 5

[2,1,5,4,3] => [1,2,1,1] => 11011 => [1,1,2,1,1] => 4

[2,3,1,4,5] => [2,3] => 10100 => [1,2,3] => 5

[2,3,1,5,4] => [2,2,1] => 10101 => [1,2,2,1] => 6

[2,3,4,1,5] => [3,2] => 10010 => [1,3,2] => 5

[2,3,4,5,1] => [4,1] => 10001 => [1,4,1] => 4

[2,3,5,1,4] => [3,2] => 10010 => [1,3,2] => 5

[2,3,5,4,1] => [3,1,1] => 10011 => [1,3,1,1] => 4

[2,4,1,3,5] => [2,3] => 10100 => [1,2,3] => 5

[2,4,1,5,3] => [2,2,1] => 10101 => [1,2,2,1] => 6

[2,4,3,1,5] => [2,1,2] => 10110 => [1,2,1,2] => 5

[2,4,3,5,1] => [2,2,1] => 10101 => [1,2,2,1] => 6

[2,4,5,1,3] => [3,2] => 10010 => [1,3,2] => 5

[2,4,5,3,1] => [3,1,1] => 10011 => [1,3,1,1] => 4

[2,5,1,3,4] => [2,3] => 10100 => [1,2,3] => 5

[2,5,1,4,3] => [2,2,1] => 10101 => [1,2,2,1] => 6

[2,5,3,1,4] => [2,1,2] => 10110 => [1,2,1,2] => 5

[2,5,3,4,1] => [2,2,1] => 10101 => [1,2,2,1] => 6

[2,5,4,1,3] => [2,1,2] => 10110 => [1,2,1,2] => 5

[2,5,4,3,1] => [2,1,1,1] => 10111 => [1,2,1,1,1] => 4

[3,1,2,4,5] => [1,4] => 11000 => [1,1,4] => 3

[3,1,2,5,4] => [1,3,1] => 11001 => [1,1,3,1] => 4

[3,1,4,2,5] => [1,2,2] => 11010 => [1,1,2,2] => 5

[3,1,4,5,2] => [1,3,1] => 11001 => [1,1,3,1] => 4

[3,1,5,2,4] => [1,2,2] => 11010 => [1,1,2,2] => 5

[3,1,5,4,2] => [1,2,1,1] => 11011 => [1,1,2,1,1] => 4

[3,2,1,4,5] => [1,1,3] => 11100 => [1,1,1,3] => 3

[3,2,1,5,4] => [1,1,2,1] => 11101 => [1,1,1,2,1] => 4

[3,2,4,1,5] => [1,2,2] => 11010 => [1,1,2,2] => 5

[3,2,4,5,1] => [1,3,1] => 11001 => [1,1,3,1] => 4

[3,2,5,1,4] => [1,2,2] => 11010 => [1,1,2,2] => 5

[3,2,5,4,1] => [1,2,1,1] => 11011 => [1,1,2,1,1] => 4

[3,4,1,2,5] => [2,3] => 10100 => [1,2,3] => 5

[3,4,1,5,2] => [2,2,1] => 10101 => [1,2,2,1] => 6

[3,4,2,1,5] => [2,1,2] => 10110 => [1,2,1,2] => 5

[3,4,2,5,1] => [2,2,1] => 10101 => [1,2,2,1] => 6

[3,4,5,1,2] => [3,2] => 10010 => [1,3,2] => 5

[3,4,5,2,1] => [3,1,1] => 10011 => [1,3,1,1] => 4

[3,5,1,2,4] => [2,3] => 10100 => [1,2,3] => 5

[3,5,1,4,2] => [2,2,1] => 10101 => [1,2,2,1] => 6

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Description

The number of corners of the ribbon associated with an integer composition.
We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell.
This statistic records the total number of corners of the ribbon shape.

Map

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.

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.

Map

to binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.