Identifier
-
Mp00023:
Dyck paths
—to non-crossing permutation⟶
Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ
Values
[1,0] => [1] => => [1] => 1
[1,0,1,0] => [1,2] => 0 => [2] => 1
[1,1,0,0] => [2,1] => 1 => [1,1] => 2
[1,0,1,0,1,0] => [1,2,3] => 00 => [3] => 1
[1,0,1,1,0,0] => [1,3,2] => 01 => [2,1] => 2
[1,1,0,0,1,0] => [2,1,3] => 10 => [1,2] => 2
[1,1,0,1,0,0] => [2,3,1] => 01 => [2,1] => 2
[1,1,1,0,0,0] => [3,2,1] => 11 => [1,1,1] => 3
[1,0,1,0,1,0,1,0] => [1,2,3,4] => 000 => [4] => 1
[1,0,1,0,1,1,0,0] => [1,2,4,3] => 001 => [3,1] => 2
[1,0,1,1,0,0,1,0] => [1,3,2,4] => 010 => [2,2] => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => 001 => [3,1] => 2
[1,0,1,1,1,0,0,0] => [1,4,3,2] => 011 => [2,1,1] => 3
[1,1,0,0,1,0,1,0] => [2,1,3,4] => 100 => [1,3] => 2
[1,1,0,0,1,1,0,0] => [2,1,4,3] => 101 => [1,2,1] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => 010 => [2,2] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => 001 => [3,1] => 2
[1,1,0,1,1,0,0,0] => [2,4,3,1] => 011 => [2,1,1] => 3
[1,1,1,0,0,0,1,0] => [3,2,1,4] => 110 => [1,1,2] => 3
[1,1,1,0,0,1,0,0] => [3,2,4,1] => 011 => [2,1,1] => 3
[1,1,1,0,1,0,0,0] => [4,2,3,1] => 011 => [2,1,1] => 3
[1,1,1,1,0,0,0,0] => [4,3,2,1] => 111 => [1,1,1,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0000 => [5] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 0001 => [4,1] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0010 => [3,2] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 0001 => [4,1] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 0011 => [3,1,1] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0100 => [2,3] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0101 => [2,2,1] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 0010 => [3,2] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 0001 => [4,1] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 0011 => [3,1,1] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 0110 => [2,1,2] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 0011 => [3,1,1] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => 0011 => [3,1,1] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 0111 => [2,1,1,1] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1000 => [1,4] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1001 => [1,3,1] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1010 => [1,2,2] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 1001 => [1,3,1] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 1011 => [1,2,1,1] => 3
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 0100 => [2,3] => 2
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 0101 => [2,2,1] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 0010 => [3,2] => 2
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0001 => [4,1] => 2
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 0011 => [3,1,1] => 3
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 0110 => [2,1,2] => 3
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 0011 => [3,1,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => 0011 => [3,1,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 0111 => [2,1,1,1] => 4
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 1100 => [1,1,3] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 1101 => [1,1,2,1] => 3
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 0110 => [2,1,2] => 3
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 0101 => [2,2,1] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 0111 => [2,1,1,1] => 4
[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => 0110 => [2,1,2] => 3
[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => 0011 => [3,1,1] => 3
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => 0011 => [3,1,1] => 3
[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => 0111 => [2,1,1,1] => 4
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 1110 => [1,1,1,2] => 4
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0111 => [2,1,1,1] => 4
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => 0111 => [2,1,1,1] => 4
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => 1011 => [1,2,1,1] => 3
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 1111 => [1,1,1,1,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 00000 => [6] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 00001 => [5,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 00010 => [4,2] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => 00001 => [5,1] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 00011 => [4,1,1] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 00100 => [3,3] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 00101 => [3,2,1] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => 00010 => [4,2] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => 00001 => [5,1] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => 00011 => [4,1,1] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 00110 => [3,1,2] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => 00011 => [4,1,1] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => 00011 => [4,1,1] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 00111 => [3,1,1,1] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 01000 => [2,4] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 01001 => [2,3,1] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 01010 => [2,2,2] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => 01001 => [2,3,1] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 01011 => [2,2,1,1] => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => 00100 => [3,3] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => 00101 => [3,2,1] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => 00010 => [4,2] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => 00001 => [5,1] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => 00011 => [4,1,1] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => 00110 => [3,1,2] => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => 00011 => [4,1,1] => 3
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => 00011 => [4,1,1] => 3
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 00111 => [3,1,1,1] => 4
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 01100 => [2,1,3] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 01101 => [2,1,2,1] => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => 00110 => [3,1,2] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => 00101 => [3,2,1] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 00111 => [3,1,1,1] => 4
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => 00110 => [3,1,2] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => 00011 => [4,1,1] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => 00011 => [4,1,1] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => 00111 => [3,1,1,1] => 4
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Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
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.
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
descent tops
Description
The descent tops of a permutation as a binary word.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.
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