- FindStat

Identifier

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤ (values match St001206The maximal dimension of an indecomposable projective $eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$ .)

Values

001 => [2,1] => [1,1] => [1,0,1,0] => 2

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

100 => [1,2] => [1,1] => [1,0,1,0] => 2

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

0001 => [3,1] => [1,1] => [1,0,1,0] => 2

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

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

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

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

1000 => [1,3] => [1,1] => [1,0,1,0] => 2

1001 => [1,2,1] => [1,1,1] => [1,0,1,0,1,0] => 2

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

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

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

00001 => [4,1] => [1,1] => [1,0,1,0] => 2

00010 => [3,1,1] => [1,2] => [1,0,1,1,0,0] => 2

00011 => [3,2] => [1,1] => [1,0,1,0] => 2

00100 => [2,1,2] => [1,1,1] => [1,0,1,0,1,0] => 2

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

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

00111 => [2,3] => [1,1] => [1,0,1,0] => 2

01000 => [1,1,3] => [2,1] => [1,1,0,0,1,0] => 2

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

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

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

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

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

01111 => [1,4] => [1,1] => [1,0,1,0] => 2

10000 => [1,4] => [1,1] => [1,0,1,0] => 2

10001 => [1,3,1] => [1,1,1] => [1,0,1,0,1,0] => 2

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

10011 => [1,2,2] => [1,2] => [1,0,1,1,0,0] => 2

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

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

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

11000 => [2,3] => [1,1] => [1,0,1,0] => 2

11001 => [2,2,1] => [2,1] => [1,1,0,0,1,0] => 2

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

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

11100 => [3,2] => [1,1] => [1,0,1,0] => 2

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

11110 => [4,1] => [1,1] => [1,0,1,0] => 2

000001 => [5,1] => [1,1] => [1,0,1,0] => 2

000010 => [4,1,1] => [1,2] => [1,0,1,1,0,0] => 2

000011 => [4,2] => [1,1] => [1,0,1,0] => 2

000100 => [3,1,2] => [1,1,1] => [1,0,1,0,1,0] => 2

000101 => [3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 2

000110 => [3,2,1] => [1,1,1] => [1,0,1,0,1,0] => 2

001000 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0] => 2

001001 => [2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 2

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

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

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

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

001111 => [2,4] => [1,1] => [1,0,1,0] => 2

010000 => [1,1,4] => [2,1] => [1,1,0,0,1,0] => 2

010001 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

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

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

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

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

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

011000 => [1,2,3] => [1,1,1] => [1,0,1,0,1,0] => 2

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

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

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

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

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

011110 => [1,4,1] => [1,1,1] => [1,0,1,0,1,0] => 2

011111 => [1,5] => [1,1] => [1,0,1,0] => 2

100000 => [1,5] => [1,1] => [1,0,1,0] => 2

100001 => [1,4,1] => [1,1,1] => [1,0,1,0,1,0] => 2

100010 => [1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0] => 2

100011 => [1,3,2] => [1,1,1] => [1,0,1,0,1,0] => 2

100100 => [1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 2

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

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

100111 => [1,2,3] => [1,1,1] => [1,0,1,0,1,0] => 2

101000 => [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0] => 2

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

101011 => [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 2

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

101101 => [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 2

101110 => [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0] => 2

101111 => [1,1,4] => [2,1] => [1,1,0,0,1,0] => 2

110000 => [2,4] => [1,1] => [1,0,1,0] => 2

110001 => [2,3,1] => [1,1,1] => [1,0,1,0,1,0] => 2

110010 => [2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0] => 2

110100 => [2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0] => 2

110101 => [2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 2

110110 => [2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 2

110111 => [2,1,3] => [1,1,1] => [1,0,1,0,1,0] => 2

111001 => [3,2,1] => [1,1,1] => [1,0,1,0,1,0] => 2

111010 => [3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0] => 2

111011 => [3,1,2] => [1,1,1] => [1,0,1,0,1,0] => 2

111100 => [4,2] => [1,1] => [1,0,1,0] => 2

111101 => [4,1,1] => [1,2] => [1,0,1,1,0,0] => 2

111110 => [5,1] => [1,1] => [1,0,1,0] => 2

0000001 => [6,1] => [1,1] => [1,0,1,0] => 2

0000010 => [5,1,1] => [1,2] => [1,0,1,1,0,0] => 2

0000011 => [5,2] => [1,1] => [1,0,1,0] => 2

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searching the database for the individual values of this statistic

Description

The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Map

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition

$C$ is the compositions composed of the lengths of consecutive runs of the same integer in

$C$ .

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

$w$ .

Map

bounce path

Description

The bounce path determined by an integer composition.