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Identifier

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤ (values match St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: )

Values

[1] => 10 => [1,2] => [1,0,1,1,0,0] => 1
[2] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1
[1,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2
[3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
[2,1] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,1] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 3
[4] => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
[3,1] => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 2
[2,2] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 2
[2,1,1] => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 3
[1,1,1,1] => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 4
[5] => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
[4,1] => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 2
[3,2] => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 2
[2,2,1] => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 3
[2,1,1,1] => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 4
[1,1,1,1,1] => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 5
[6] => 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 1
[5,1] => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2
[4,2] => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 2
[3,3] => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[2,2,2] => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[2,2,1,1] => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 4
[2,1,1,1,1] => 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 5
[1,1,1,1,1,1] => 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 6
[7] => 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 1
[6,1] => 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 2
[4,3] => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 2
[3,3,1] => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 3
[2,2,2,1] => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 4
[2,2,1,1,1] => 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 5
[2,1,1,1,1,1] => 10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 6
[1,1,1,1,1,1,1] => 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 7
[8] => 100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 1
[5,3] => 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 2
[4,4] => 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 2
[2,2,2,1,1] => 1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 5
[1,1,1,1,1,1,1,1] => 111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 8
[9] => 1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => 1
[5,4] => 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 2
[2,2,2,2,1] => 1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 5
[1,1,1,1,1,1,1,1,1] => 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 9
[5,5] => 1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 2
[6,5] => 10100000 => [1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 2
[2,2,2,2,2,1] => 11111010 => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 6
[6,6] => 11000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 2
[7,7] => 110000000 => [1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 2
[] =>  => [1] => [1,0] => 0

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + q^{2} + q^{3}$

$F_{4} = q + 2\ q^{2} + q^{3} + q^{4}$

Description

The bounce count of a Dyck path.
For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the bounce path of

$D$ .

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 partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.

Map

bounce path

Description

The bounce path determined by an integer composition.