- FindStat

Your data matches 182 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000808
(load all 4 compositions to match this statistic)

Mapping

St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1 = 0 + 1
[1,1] => 1 = 0 + 1
[2] => 2 = 1 + 1
[1,1,1] => 1 = 0 + 1
[1,2] => 2 = 1 + 1
[2,1] => 2 = 1 + 1
[3] => 3 = 2 + 1
[1,1,1,1] => 1 = 0 + 1
[1,1,2] => 2 = 1 + 1
[1,2,1] => 2 = 1 + 1
[1,3] => 3 = 2 + 1
[2,1,1] => 2 = 1 + 1
[2,2] => 2 = 1 + 1
[3,1] => 3 = 2 + 1
[4] => 4 = 3 + 1
[1,1,1,1,1] => 1 = 0 + 1
[1,1,1,2] => 2 = 1 + 1
[1,1,2,1] => 2 = 1 + 1
[1,1,3] => 3 = 2 + 1
[1,2,1,1] => 2 = 1 + 1
[1,2,2] => 2 = 1 + 1
[1,3,1] => 3 = 2 + 1
[1,4] => 4 = 3 + 1
[2,1,1,1] => 2 = 1 + 1
[2,1,2] => 3 = 2 + 1
[2,2,1] => 2 = 1 + 1
[2,3] => 3 = 2 + 1
[3,1,1] => 3 = 2 + 1
[3,2] => 3 = 2 + 1
[4,1] => 4 = 3 + 1
[5] => 5 = 4 + 1
[1,1,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,2] => 2 = 1 + 1
[1,1,1,2,1] => 2 = 1 + 1
[1,1,1,3] => 3 = 2 + 1
[1,1,2,1,1] => 2 = 1 + 1
[1,1,2,2] => 2 = 1 + 1
[1,1,3,1] => 3 = 2 + 1
[1,1,4] => 4 = 3 + 1
[1,2,1,1,1] => 2 = 1 + 1
[1,2,1,2] => 3 = 2 + 1
[1,2,2,1] => 2 = 1 + 1
[1,2,3] => 3 = 2 + 1
[1,3,1,1] => 3 = 2 + 1
[1,3,2] => 3 = 2 + 1
[1,4,1] => 4 = 3 + 1
[1,5] => 5 = 4 + 1
[2,1,1,1,1] => 2 = 1 + 1
[2,1,1,2] => 3 = 2 + 1
[2,1,2,1] => 3 = 2 + 1

Description

The number of up steps of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.

Matching statistic: St000335
(load all 5 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => 1 = 0 + 1
[1,1] => [1,0,1,0]
 => 1 = 0 + 1
[2] => [1,1,0,0]
 => 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3] => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[4] => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1

Description

The difference of lower and upper interactions. An ''upper interaction'' in a Dyck path is the occurrence of a factor

$0^k 1^k$ with

$k \geq 1$ (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor

$1^k 0^k$ with

$k \geq 1$ . In both cases,

$1$ denotes an up-step

$0$ denotes a a down-step.

Matching statistic: St000091
(load all 2 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000091: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1 => [1,1] => 0
[1,1] => 11 => [1,1,1] => 0
[2] => 10 => [1,2] => 1
[1,1,1] => 111 => [1,1,1,1] => 0
[1,2] => 110 => [1,1,2] => 1
[2,1] => 101 => [1,2,1] => 1
[3] => 100 => [1,3] => 2
[1,1,1,1] => 1111 => [1,1,1,1,1] => 0
[1,1,2] => 1110 => [1,1,1,2] => 1
[1,2,1] => 1101 => [1,1,2,1] => 1
[1,3] => 1100 => [1,1,3] => 2
[2,1,1] => 1011 => [1,2,1,1] => 1
[2,2] => 1010 => [1,2,2] => 1
[3,1] => 1001 => [1,3,1] => 2
[4] => 1000 => [1,4] => 3
[1,1,1,1,1] => 11111 => [1,1,1,1,1,1] => 0
[1,1,1,2] => 11110 => [1,1,1,1,2] => 1
[1,1,2,1] => 11101 => [1,1,1,2,1] => 1
[1,1,3] => 11100 => [1,1,1,3] => 2
[1,2,1,1] => 11011 => [1,1,2,1,1] => 1
[1,2,2] => 11010 => [1,1,2,2] => 1
[1,3,1] => 11001 => [1,1,3,1] => 2
[1,4] => 11000 => [1,1,4] => 3
[2,1,1,1] => 10111 => [1,2,1,1,1] => 1
[2,1,2] => 10110 => [1,2,1,2] => 2
[2,2,1] => 10101 => [1,2,2,1] => 1
[2,3] => 10100 => [1,2,3] => 2
[3,1,1] => 10011 => [1,3,1,1] => 2
[3,2] => 10010 => [1,3,2] => 2
[4,1] => 10001 => [1,4,1] => 3
[5] => 10000 => [1,5] => 4
[1,1,1,1,1,1] => 111111 => [1,1,1,1,1,1,1] => 0
[1,1,1,1,2] => 111110 => [1,1,1,1,1,2] => 1
[1,1,1,2,1] => 111101 => [1,1,1,1,2,1] => 1
[1,1,1,3] => 111100 => [1,1,1,1,3] => 2
[1,1,2,1,1] => 111011 => [1,1,1,2,1,1] => 1
[1,1,2,2] => 111010 => [1,1,1,2,2] => 1
[1,1,3,1] => 111001 => [1,1,1,3,1] => 2
[1,1,4] => 111000 => [1,1,1,4] => 3
[1,2,1,1,1] => 110111 => [1,1,2,1,1,1] => 1
[1,2,1,2] => 110110 => [1,1,2,1,2] => 2
[1,2,2,1] => 110101 => [1,1,2,2,1] => 1
[1,2,3] => 110100 => [1,1,2,3] => 2
[1,3,1,1] => 110011 => [1,1,3,1,1] => 2
[1,3,2] => 110010 => [1,1,3,2] => 2
[1,4,1] => 110001 => [1,1,4,1] => 3
[1,5] => 110000 => [1,1,5] => 4
[2,1,1,1,1] => 101111 => [1,2,1,1,1,1] => 1
[2,1,1,2] => 101110 => [1,2,1,1,2] => 2
[2,1,2,1] => 101101 => [1,2,1,2,1] => 2

Description

The descent variation of a composition. Defined in [1].

Matching statistic: St001372
(load all 2 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1 => 0 => 0
[1,1] => 11 => 00 => 0
[2] => 10 => 01 => 1
[1,1,1] => 111 => 000 => 0
[1,2] => 110 => 001 => 1
[2,1] => 101 => 010 => 1
[3] => 100 => 011 => 2
[1,1,1,1] => 1111 => 0000 => 0
[1,1,2] => 1110 => 0001 => 1
[1,2,1] => 1101 => 0010 => 1
[1,3] => 1100 => 0011 => 2
[2,1,1] => 1011 => 0100 => 1
[2,2] => 1010 => 0101 => 1
[3,1] => 1001 => 0110 => 2
[4] => 1000 => 0111 => 3
[1,1,1,1,1] => 11111 => 00000 => 0
[1,1,1,2] => 11110 => 00001 => 1
[1,1,2,1] => 11101 => 00010 => 1
[1,1,3] => 11100 => 00011 => 2
[1,2,1,1] => 11011 => 00100 => 1
[1,2,2] => 11010 => 00101 => 1
[1,3,1] => 11001 => 00110 => 2
[1,4] => 11000 => 00111 => 3
[2,1,1,1] => 10111 => 01000 => 1
[2,1,2] => 10110 => 01001 => 1
[2,2,1] => 10101 => 01010 => 2
[2,3] => 10100 => 01011 => 2
[3,1,1] => 10011 => 01100 => 2
[3,2] => 10010 => 01101 => 2
[4,1] => 10001 => 01110 => 3
[5] => 10000 => 01111 => 4
[1,1,1,1,1,1] => 111111 => 000000 => 0
[1,1,1,1,2] => 111110 => 000001 => 1
[1,1,1,2,1] => 111101 => 000010 => 1
[1,1,1,3] => 111100 => 000011 => 2
[1,1,2,1,1] => 111011 => 000100 => 1
[1,1,2,2] => 111010 => 000101 => 1
[1,1,3,1] => 111001 => 000110 => 2
[1,1,4] => 111000 => 000111 => 3
[1,2,1,1,1] => 110111 => 001000 => 1
[1,2,1,2] => 110110 => 001001 => 1
[1,2,2,1] => 110101 => 001010 => 2
[1,2,3] => 110100 => 001011 => 2
[1,3,1,1] => 110011 => 001100 => 2
[1,3,2] => 110010 => 001101 => 2
[1,4,1] => 110001 => 001110 => 3
[1,5] => 110000 => 001111 => 4
[2,1,1,1,1] => 101111 => 010000 => 1
[2,1,1,2] => 101110 => 010001 => 1
[2,1,2,1] => 101101 => 010010 => 2

Description

The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.

Matching statistic: St000120
(load all 3 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[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]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 3
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 3

Description

The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b

Matching statistic: St000864
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1,0]
 => [1] => 0
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 4
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,4,3,1,6,5] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,6,5,3] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,4,6,3,1] => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,4,3,1,5,6] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,1,4,5,6,3] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,4,3,1] => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,1,5,4,6,3] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,5,3,6,1] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,6,5,4,3] => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,1,5,6,4,3] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,5,6,3,1] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [3,2,4,1,6,5] => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,6,5] => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,2,4,6,5,1] => 2

Description

The number of circled entries of the shifted recording tableau of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the number of circled entries in

$Q$ .

Matching statistic: St001189

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 2
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 2

Description

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001192

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 0
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 3
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 2

Description

The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ .

Matching statistic: St001205
(load all 6 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001205: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[2] => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 3
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 2

Description

The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Nakayama algebra and the relation to Dyck paths.

Matching statistic: St000013
(load all 2 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,1] => [1,0,1,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[2] => [1,1,0,0]
 => [1,1,0,0]
 => [1,1,0,0]
 => 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 4 = 3 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 4 = 3 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 4 = 3 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 4 = 3 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

The following 172 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000686The finitistic dominant dimension of a Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000442The maximal area to the right of an up step of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000702The number of weak deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000306The bounce count of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000444The length of the maximal rise of a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000982The length of the longest constant subword. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000681The Grundy value of Chomp on Ferrers diagrams. St001644The dimension of a graph. St000451The length of the longest pattern of the form k 1 2. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001645The pebbling number of a connected graph. St000456The monochromatic index of a connected graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000675The number of centered multitunnels of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000939The number of characters of the symmetric group whose value on the partition is positive. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001933The largest multiplicity of a part in an integer partition. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001118The acyclic chromatic index of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition. St000993The multiplicity of the largest part of an integer partition. St001060The distinguishing index of a graph. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001414Half the length of the longest odd length palindromic prefix of a binary word. St000455The second largest eigenvalue of a graph if it is integral. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001863The number of weak excedances of a signed permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000667The greatest common divisor of the parts of the partition. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001557The number of inversions of the second entry of a permutation. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001712The number of natural descents of a standard Young tableau. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001948The number of augmented double ascents of a permutation. St000983The length of the longest alternating subword. St000381The largest part of an integer composition. St001267The length of the Lyndon factorization of the binary word. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St000871The number of very big ascents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000141The maximum drop size of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000632The jump number of the poset. St000307The number of rowmotion orbits of a poset. St000662The staircase size of the code of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000023The number of inner peaks of a permutation. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000234The number of global ascents of a permutation. St000260The radius of a connected graph. St000264The girth of a graph, which is not a tree. St000353The number of inner valleys of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001728The number of invisible descents of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000239The number of small weak excedances. St000254The nesting number of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001864The number of excedances of a signed permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.