- FindStat

Identifier

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤ

Values

0 => [2] => [2] => [1,1,0,0,1,0] => 3

1 => [1,1] => [1,1] => [1,0,1,1,0,0] => 2

00 => [3] => [3] => [1,1,1,0,0,0,1,0] => 4

01 => [2,1] => [2,1] => [1,0,1,0,1,0] => 3

10 => [1,2] => [2,1] => [1,0,1,0,1,0] => 3

11 => [1,1,1] => [1,1,1] => [1,0,1,1,1,0,0,0] => 2

000 => [4] => [4] => [1,1,1,1,0,0,0,0,1,0] => 5

001 => [3,1] => [3,1] => [1,1,0,1,0,0,1,0] => 4

010 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 3

011 => [2,1,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3

100 => [1,3] => [3,1] => [1,1,0,1,0,0,1,0] => 4

101 => [1,2,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3

110 => [1,1,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => 3

111 => [1,1,1,1] => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => 2

0000 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 6

0001 => [4,1] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 5

0010 => [3,2] => [3,2] => [1,1,0,0,1,0,1,0] => 4

0011 => [3,1,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => 4

0100 => [2,3] => [3,2] => [1,1,0,0,1,0,1,0] => 4

0101 => [2,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => 3

0110 => [2,1,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => 3

0111 => [2,1,1,1] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => 3

1000 => [1,4] => [4,1] => [1,1,1,0,1,0,0,0,1,0] => 5

1001 => [1,3,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => 4

1010 => [1,2,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => 3

1011 => [1,2,1,1] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => 3

1100 => [1,1,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => 4

1101 => [1,1,2,1] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => 3

1110 => [1,1,1,2] => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => 3

1111 => [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2

00001 => [5,1] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 6

00010 => [4,2] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 5

00011 => [4,1,1] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5

00100 => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,0,0] => 4

00101 => [3,2,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

00110 => [3,1,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

00111 => [3,1,1,1] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => 4

01000 => [2,4] => [4,2] => [1,1,1,0,0,1,0,0,1,0] => 5

01001 => [2,3,1] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

01010 => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => 3

01011 => [2,2,1,1] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 3

01100 => [2,1,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

01101 => [2,1,2,1] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 3

01110 => [2,1,1,2] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 3

01111 => [2,1,1,1,1] => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3

10000 => [1,5] => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => 6

10001 => [1,4,1] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5

10010 => [1,3,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

10011 => [1,3,1,1] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => 4

10100 => [1,2,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => 4

10101 => [1,2,2,1] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 3

10110 => [1,2,1,2] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 3

10111 => [1,2,1,1,1] => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3

11000 => [1,1,4] => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5

11001 => [1,1,3,1] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => 4

11010 => [1,1,2,2] => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => 3

11011 => [1,1,2,1,1] => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3

11100 => [1,1,1,3] => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => 4

11101 => [1,1,1,2,1] => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3

11110 => [1,1,1,1,2] => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => 3

000010 => [5,2] => [5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => 6

000011 => [5,1,1] => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 6

000100 => [4,3] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 5

000101 => [4,2,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 5

000110 => [4,1,2] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 5

000111 => [4,1,1,1] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => 5

001000 => [3,4] => [4,3] => [1,1,1,0,0,0,1,0,1,0] => 5

001001 => [3,3,1] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => 4

001010 => [3,2,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => 4

001011 => [3,2,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

001100 => [3,1,3] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => 4

001101 => [3,1,2,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

001110 => [3,1,1,2] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

001111 => [3,1,1,1,1] => [3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4

010000 => [2,5] => [5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => 6

010001 => [2,4,1] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 5

010010 => [2,3,2] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => 4

010011 => [2,3,1,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

010100 => [2,2,3] => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => 4

010101 => [2,2,2,1] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => 3

010110 => [2,2,1,2] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => 3

010111 => [2,2,1,1,1] => [2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => 3

011000 => [2,1,4] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 5

011001 => [2,1,3,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

011010 => [2,1,2,2] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => 3

011011 => [2,1,2,1,1] => [2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => 3

011100 => [2,1,1,3] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

011101 => [2,1,1,2,1] => [2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => 3

011110 => [2,1,1,1,2] => [2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => 3

100001 => [1,5,1] => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 6

100010 => [1,4,2] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 5

100011 => [1,4,1,1] => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => 5

100100 => [1,3,3] => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => 4

100101 => [1,3,2,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

100110 => [1,3,1,2] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

100111 => [1,3,1,1,1] => [3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => 4

101000 => [1,2,4] => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => 5

101001 => [1,2,3,1] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

101010 => [1,2,2,2] => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => 3

101011 => [1,2,2,1,1] => [2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => 3

101100 => [1,2,1,3] => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => 4

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Description

The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in DyckPaths/NakayamaAlgebras. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

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.