Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001195: Dyck paths ⟶ ℤ
Values
000 => [3] => [1,1,1,0,0,0] => [1,1,0,0,1,0] => 0
001 => [2,1] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 1
010 => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 1
011 => [1,2] => [1,0,1,1,0,0] => [1,1,1,0,0,0] => 1
100 => [1,2] => [1,0,1,1,0,0] => [1,1,1,0,0,0] => 1
101 => [1,1,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 1
110 => [2,1] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 1
111 => [3] => [1,1,1,0,0,0] => [1,1,0,0,1,0] => 0
0000 => [4] => [1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => 1
0001 => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
0111 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
1111 => [4] => [1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 1
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 1
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 1
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
10111 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 1
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => 1
11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => 1
11011 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => 1
11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => 1
11101 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => 1
11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
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Description
The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word w is the integer compositions composed of the lengths of consecutive runs of the same letter in w.
The delta morphism of a finite word w is the integer compositions composed of the lengths of consecutive runs of the same letter in w.
Map
inverse promotion
Description
The inverse promotion of a Dyck path.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.
This is the bijection obtained by applying the inverse of Schützenberger's promotion to the corresponding two rowed standard Young tableau.
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