Identifier
-
Mp00102:
Dyck paths
—rise composition⟶
Integer compositions
St001235: Integer compositions ⟶ ℤ
Values
=>
Cc0005;cc-rep-0
[1,0]=>[1]=>1
[1,0,1,0]=>[1,1]=>2
[1,1,0,0]=>[2]=>1
[1,0,1,0,1,0]=>[1,1,1]=>3
[1,0,1,1,0,0]=>[1,2]=>2
[1,1,0,0,1,0]=>[2,1]=>2
[1,1,0,1,0,0]=>[2,1]=>2
[1,1,1,0,0,0]=>[3]=>1
[1,0,1,0,1,0,1,0]=>[1,1,1,1]=>4
[1,0,1,0,1,1,0,0]=>[1,1,2]=>3
[1,0,1,1,0,0,1,0]=>[1,2,1]=>2
[1,0,1,1,0,1,0,0]=>[1,2,1]=>2
[1,0,1,1,1,0,0,0]=>[1,3]=>2
[1,1,0,0,1,0,1,0]=>[2,1,1]=>3
[1,1,0,0,1,1,0,0]=>[2,2]=>2
[1,1,0,1,0,0,1,0]=>[2,1,1]=>3
[1,1,0,1,0,1,0,0]=>[2,1,1]=>3
[1,1,0,1,1,0,0,0]=>[2,2]=>2
[1,1,1,0,0,0,1,0]=>[3,1]=>2
[1,1,1,0,0,1,0,0]=>[3,1]=>2
[1,1,1,0,1,0,0,0]=>[3,1]=>2
[1,1,1,1,0,0,0,0]=>[4]=>1
[1,0,1,0,1,0,1,0,1,0]=>[1,1,1,1,1]=>5
[1,0,1,0,1,0,1,1,0,0]=>[1,1,1,2]=>4
[1,0,1,0,1,1,0,0,1,0]=>[1,1,2,1]=>3
[1,0,1,0,1,1,0,1,0,0]=>[1,1,2,1]=>3
[1,0,1,0,1,1,1,0,0,0]=>[1,1,3]=>3
[1,0,1,1,0,0,1,0,1,0]=>[1,2,1,1]=>3
[1,0,1,1,0,0,1,1,0,0]=>[1,2,2]=>2
[1,0,1,1,0,1,0,0,1,0]=>[1,2,1,1]=>3
[1,0,1,1,0,1,0,1,0,0]=>[1,2,1,1]=>3
[1,0,1,1,0,1,1,0,0,0]=>[1,2,2]=>2
[1,0,1,1,1,0,0,0,1,0]=>[1,3,1]=>2
[1,0,1,1,1,0,0,1,0,0]=>[1,3,1]=>2
[1,0,1,1,1,0,1,0,0,0]=>[1,3,1]=>2
[1,0,1,1,1,1,0,0,0,0]=>[1,4]=>2
[1,1,0,0,1,0,1,0,1,0]=>[2,1,1,1]=>4
[1,1,0,0,1,0,1,1,0,0]=>[2,1,2]=>3
[1,1,0,0,1,1,0,0,1,0]=>[2,2,1]=>2
[1,1,0,0,1,1,0,1,0,0]=>[2,2,1]=>2
[1,1,0,0,1,1,1,0,0,0]=>[2,3]=>2
[1,1,0,1,0,0,1,0,1,0]=>[2,1,1,1]=>4
[1,1,0,1,0,0,1,1,0,0]=>[2,1,2]=>3
[1,1,0,1,0,1,0,0,1,0]=>[2,1,1,1]=>4
[1,1,0,1,0,1,0,1,0,0]=>[2,1,1,1]=>4
[1,1,0,1,0,1,1,0,0,0]=>[2,1,2]=>3
[1,1,0,1,1,0,0,0,1,0]=>[2,2,1]=>2
[1,1,0,1,1,0,0,1,0,0]=>[2,2,1]=>2
[1,1,0,1,1,0,1,0,0,0]=>[2,2,1]=>2
[1,1,0,1,1,1,0,0,0,0]=>[2,3]=>2
[1,1,1,0,0,0,1,0,1,0]=>[3,1,1]=>3
[1,1,1,0,0,0,1,1,0,0]=>[3,2]=>2
[1,1,1,0,0,1,0,0,1,0]=>[3,1,1]=>3
[1,1,1,0,0,1,0,1,0,0]=>[3,1,1]=>3
[1,1,1,0,0,1,1,0,0,0]=>[3,2]=>2
[1,1,1,0,1,0,0,0,1,0]=>[3,1,1]=>3
[1,1,1,0,1,0,0,1,0,0]=>[3,1,1]=>3
[1,1,1,0,1,0,1,0,0,0]=>[3,1,1]=>3
[1,1,1,0,1,1,0,0,0,0]=>[3,2]=>2
[1,1,1,1,0,0,0,0,1,0]=>[4,1]=>2
[1,1,1,1,0,0,0,1,0,0]=>[4,1]=>2
[1,1,1,1,0,0,1,0,0,0]=>[4,1]=>2
[1,1,1,1,0,1,0,0,0,0]=>[4,1]=>2
[1,1,1,1,1,0,0,0,0,0]=>[5]=>1
[1,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,1,1,1,1]=>6
[1,0,1,0,1,0,1,0,1,1,0,0]=>[1,1,1,1,2]=>5
[1,0,1,0,1,0,1,1,0,0,1,0]=>[1,1,1,2,1]=>4
[1,0,1,0,1,0,1,1,0,1,0,0]=>[1,1,1,2,1]=>4
[1,0,1,0,1,0,1,1,1,0,0,0]=>[1,1,1,3]=>4
[1,0,1,0,1,1,0,0,1,0,1,0]=>[1,1,2,1,1]=>3
[1,0,1,0,1,1,0,0,1,1,0,0]=>[1,1,2,2]=>3
[1,0,1,0,1,1,0,1,0,0,1,0]=>[1,1,2,1,1]=>3
[1,0,1,0,1,1,0,1,0,1,0,0]=>[1,1,2,1,1]=>3
[1,0,1,0,1,1,0,1,1,0,0,0]=>[1,1,2,2]=>3
[1,0,1,0,1,1,1,0,0,0,1,0]=>[1,1,3,1]=>3
[1,0,1,0,1,1,1,0,0,1,0,0]=>[1,1,3,1]=>3
[1,0,1,0,1,1,1,0,1,0,0,0]=>[1,1,3,1]=>3
[1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,4]=>3
[1,0,1,1,0,0,1,0,1,0,1,0]=>[1,2,1,1,1]=>4
[1,0,1,1,0,0,1,0,1,1,0,0]=>[1,2,1,2]=>3
[1,0,1,1,0,0,1,1,0,0,1,0]=>[1,2,2,1]=>2
[1,0,1,1,0,0,1,1,0,1,0,0]=>[1,2,2,1]=>2
[1,0,1,1,0,0,1,1,1,0,0,0]=>[1,2,3]=>2
[1,0,1,1,0,1,0,0,1,0,1,0]=>[1,2,1,1,1]=>4
[1,0,1,1,0,1,0,0,1,1,0,0]=>[1,2,1,2]=>3
[1,0,1,1,0,1,0,1,0,0,1,0]=>[1,2,1,1,1]=>4
[1,0,1,1,0,1,0,1,0,1,0,0]=>[1,2,1,1,1]=>4
[1,0,1,1,0,1,0,1,1,0,0,0]=>[1,2,1,2]=>3
[1,0,1,1,0,1,1,0,0,0,1,0]=>[1,2,2,1]=>2
[1,0,1,1,0,1,1,0,0,1,0,0]=>[1,2,2,1]=>2
[1,0,1,1,0,1,1,0,1,0,0,0]=>[1,2,2,1]=>2
[1,0,1,1,0,1,1,1,0,0,0,0]=>[1,2,3]=>2
[1,0,1,1,1,0,0,0,1,0,1,0]=>[1,3,1,1]=>3
[1,0,1,1,1,0,0,0,1,1,0,0]=>[1,3,2]=>2
[1,0,1,1,1,0,0,1,0,0,1,0]=>[1,3,1,1]=>3
[1,0,1,1,1,0,0,1,0,1,0,0]=>[1,3,1,1]=>3
[1,0,1,1,1,0,0,1,1,0,0,0]=>[1,3,2]=>2
[1,0,1,1,1,0,1,0,0,0,1,0]=>[1,3,1,1]=>3
[1,0,1,1,1,0,1,0,0,1,0,0]=>[1,3,1,1]=>3
[1,0,1,1,1,0,1,0,1,0,0,0]=>[1,3,1,1]=>3
[1,0,1,1,1,0,1,1,0,0,0,0]=>[1,3,2]=>2
[1,0,1,1,1,1,0,0,0,0,1,0]=>[1,4,1]=>2
[1,0,1,1,1,1,0,0,0,1,0,0]=>[1,4,1]=>2
[1,0,1,1,1,1,0,0,1,0,0,0]=>[1,4,1]=>2
[1,0,1,1,1,1,0,1,0,0,0,0]=>[1,4,1]=>2
[1,0,1,1,1,1,1,0,0,0,0,0]=>[1,5]=>2
[1,1,0,0,1,0,1,0,1,0,1,0]=>[2,1,1,1,1]=>5
[1,1,0,0,1,0,1,0,1,1,0,0]=>[2,1,1,2]=>4
[1,1,0,0,1,0,1,1,0,0,1,0]=>[2,1,2,1]=>3
[1,1,0,0,1,0,1,1,0,1,0,0]=>[2,1,2,1]=>3
[1,1,0,0,1,0,1,1,1,0,0,0]=>[2,1,3]=>3
[1,1,0,0,1,1,0,0,1,0,1,0]=>[2,2,1,1]=>3
[1,1,0,0,1,1,0,0,1,1,0,0]=>[2,2,2]=>2
[1,1,0,0,1,1,0,1,0,0,1,0]=>[2,2,1,1]=>3
[1,1,0,0,1,1,0,1,0,1,0,0]=>[2,2,1,1]=>3
[1,1,0,0,1,1,0,1,1,0,0,0]=>[2,2,2]=>2
[1,1,0,0,1,1,1,0,0,0,1,0]=>[2,3,1]=>2
[1,1,0,0,1,1,1,0,0,1,0,0]=>[2,3,1]=>2
[1,1,0,0,1,1,1,0,1,0,0,0]=>[2,3,1]=>2
[1,1,0,0,1,1,1,1,0,0,0,0]=>[2,4]=>2
[1,1,0,1,0,0,1,0,1,0,1,0]=>[2,1,1,1,1]=>5
[1,1,0,1,0,0,1,0,1,1,0,0]=>[2,1,1,2]=>4
[1,1,0,1,0,0,1,1,0,0,1,0]=>[2,1,2,1]=>3
[1,1,0,1,0,0,1,1,0,1,0,0]=>[2,1,2,1]=>3
[1,1,0,1,0,0,1,1,1,0,0,0]=>[2,1,3]=>3
[1,1,0,1,0,1,0,0,1,0,1,0]=>[2,1,1,1,1]=>5
[1,1,0,1,0,1,0,0,1,1,0,0]=>[2,1,1,2]=>4
[1,1,0,1,0,1,0,1,0,0,1,0]=>[2,1,1,1,1]=>5
[1,1,0,1,0,1,0,1,0,1,0,0]=>[2,1,1,1,1]=>5
[1,1,0,1,0,1,0,1,1,0,0,0]=>[2,1,1,2]=>4
[1,1,0,1,0,1,1,0,0,0,1,0]=>[2,1,2,1]=>3
[1,1,0,1,0,1,1,0,0,1,0,0]=>[2,1,2,1]=>3
[1,1,0,1,0,1,1,0,1,0,0,0]=>[2,1,2,1]=>3
[1,1,0,1,0,1,1,1,0,0,0,0]=>[2,1,3]=>3
[1,1,0,1,1,0,0,0,1,0,1,0]=>[2,2,1,1]=>3
[1,1,0,1,1,0,0,0,1,1,0,0]=>[2,2,2]=>2
[1,1,0,1,1,0,0,1,0,0,1,0]=>[2,2,1,1]=>3
[1,1,0,1,1,0,0,1,0,1,0,0]=>[2,2,1,1]=>3
[1,1,0,1,1,0,0,1,1,0,0,0]=>[2,2,2]=>2
[1,1,0,1,1,0,1,0,0,0,1,0]=>[2,2,1,1]=>3
[1,1,0,1,1,0,1,0,0,1,0,0]=>[2,2,1,1]=>3
[1,1,0,1,1,0,1,0,1,0,0,0]=>[2,2,1,1]=>3
[1,1,0,1,1,0,1,1,0,0,0,0]=>[2,2,2]=>2
[1,1,0,1,1,1,0,0,0,0,1,0]=>[2,3,1]=>2
[1,1,0,1,1,1,0,0,0,1,0,0]=>[2,3,1]=>2
[1,1,0,1,1,1,0,0,1,0,0,0]=>[2,3,1]=>2
[1,1,0,1,1,1,0,1,0,0,0,0]=>[2,3,1]=>2
[1,1,0,1,1,1,1,0,0,0,0,0]=>[2,4]=>2
[1,1,1,0,0,0,1,0,1,0,1,0]=>[3,1,1,1]=>4
[1,1,1,0,0,0,1,0,1,1,0,0]=>[3,1,2]=>3
[1,1,1,0,0,0,1,1,0,0,1,0]=>[3,2,1]=>2
[1,1,1,0,0,0,1,1,0,1,0,0]=>[3,2,1]=>2
[1,1,1,0,0,0,1,1,1,0,0,0]=>[3,3]=>2
[1,1,1,0,0,1,0,0,1,0,1,0]=>[3,1,1,1]=>4
[1,1,1,0,0,1,0,0,1,1,0,0]=>[3,1,2]=>3
[1,1,1,0,0,1,0,1,0,0,1,0]=>[3,1,1,1]=>4
[1,1,1,0,0,1,0,1,0,1,0,0]=>[3,1,1,1]=>4
[1,1,1,0,0,1,0,1,1,0,0,0]=>[3,1,2]=>3
[1,1,1,0,0,1,1,0,0,0,1,0]=>[3,2,1]=>2
[1,1,1,0,0,1,1,0,0,1,0,0]=>[3,2,1]=>2
[1,1,1,0,0,1,1,0,1,0,0,0]=>[3,2,1]=>2
[1,1,1,0,0,1,1,1,0,0,0,0]=>[3,3]=>2
[1,1,1,0,1,0,0,0,1,0,1,0]=>[3,1,1,1]=>4
[1,1,1,0,1,0,0,0,1,1,0,0]=>[3,1,2]=>3
[1,1,1,0,1,0,0,1,0,0,1,0]=>[3,1,1,1]=>4
[1,1,1,0,1,0,0,1,0,1,0,0]=>[3,1,1,1]=>4
[1,1,1,0,1,0,0,1,1,0,0,0]=>[3,1,2]=>3
[1,1,1,0,1,0,1,0,0,0,1,0]=>[3,1,1,1]=>4
[1,1,1,0,1,0,1,0,0,1,0,0]=>[3,1,1,1]=>4
[1,1,1,0,1,0,1,0,1,0,0,0]=>[3,1,1,1]=>4
[1,1,1,0,1,0,1,1,0,0,0,0]=>[3,1,2]=>3
[1,1,1,0,1,1,0,0,0,0,1,0]=>[3,2,1]=>2
[1,1,1,0,1,1,0,0,0,1,0,0]=>[3,2,1]=>2
[1,1,1,0,1,1,0,0,1,0,0,0]=>[3,2,1]=>2
[1,1,1,0,1,1,0,1,0,0,0,0]=>[3,2,1]=>2
[1,1,1,0,1,1,1,0,0,0,0,0]=>[3,3]=>2
[1,1,1,1,0,0,0,0,1,0,1,0]=>[4,1,1]=>3
[1,1,1,1,0,0,0,0,1,1,0,0]=>[4,2]=>2
[1,1,1,1,0,0,0,1,0,0,1,0]=>[4,1,1]=>3
[1,1,1,1,0,0,0,1,0,1,0,0]=>[4,1,1]=>3
[1,1,1,1,0,0,0,1,1,0,0,0]=>[4,2]=>2
[1,1,1,1,0,0,1,0,0,0,1,0]=>[4,1,1]=>3
[1,1,1,1,0,0,1,0,0,1,0,0]=>[4,1,1]=>3
[1,1,1,1,0,0,1,0,1,0,0,0]=>[4,1,1]=>3
[1,1,1,1,0,0,1,1,0,0,0,0]=>[4,2]=>2
[1,1,1,1,0,1,0,0,0,0,1,0]=>[4,1,1]=>3
[1,1,1,1,0,1,0,0,0,1,0,0]=>[4,1,1]=>3
[1,1,1,1,0,1,0,0,1,0,0,0]=>[4,1,1]=>3
[1,1,1,1,0,1,0,1,0,0,0,0]=>[4,1,1]=>3
[1,1,1,1,0,1,1,0,0,0,0,0]=>[4,2]=>2
[1,1,1,1,1,0,0,0,0,0,1,0]=>[5,1]=>2
[1,1,1,1,1,0,0,0,0,1,0,0]=>[5,1]=>2
[1,1,1,1,1,0,0,0,1,0,0,0]=>[5,1]=>2
[1,1,1,1,1,0,0,1,0,0,0,0]=>[5,1]=>2
[1,1,1,1,1,0,1,0,0,0,0,0]=>[5,1]=>2
[1,1,1,1,1,1,0,0,0,0,0,0]=>[6]=>1
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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
rise composition
Description
Send a Dyck path to the composition of sizes of its rises.
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