Your data matches 207 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000390
(load all 2 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 01 => 01 => 1
[1,0,1,0]
 => 1010 => 0011 => 0011 => 1
[1,1,0,0]
 => 1100 => 0011 => 0011 => 1
[1,0,1,0,1,0]
 => 101010 => 001011 => 100011 => 2
[1,0,1,1,0,0]
 => 101100 => 000111 => 000111 => 1
[1,1,0,0,1,0]
 => 110010 => 000111 => 000111 => 1
[1,1,0,1,0,0]
 => 110100 => 000111 => 000111 => 1
[1,1,1,0,0,0]
 => 111000 => 000111 => 000111 => 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 00101011 => 11000011 => 2
[1,0,1,0,1,1,0,0]
 => 10101100 => 00010111 => 10000111 => 2
[1,0,1,1,0,0,1,0]
 => 10110010 => 00010111 => 10000111 => 2
[1,0,1,1,0,1,0,0]
 => 10110100 => 00010111 => 10000111 => 2
[1,0,1,1,1,0,0,0]
 => 10111000 => 00001111 => 00001111 => 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 00010111 => 10000111 => 2
[1,1,0,0,1,1,0,0]
 => 11001100 => 00001111 => 00001111 => 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 00010111 => 10000111 => 2
[1,1,0,1,0,1,0,0]
 => 11010100 => 00010111 => 10000111 => 2
[1,1,0,1,1,0,0,0]
 => 11011000 => 00001111 => 00001111 => 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 00001111 => 00001111 => 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 00001111 => 00001111 => 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 00001111 => 00001111 => 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 00001111 => 00001111 => 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0000011111 => 0000011111 => 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0001001111 => 0100001111 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0000011111 => 0000011111 => 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0000011111 => 0000011111 => 1
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0000011111 => 0000011111 => 1
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0001001111 => 0100001111 => 2
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0000011111 => 0000011111 => 1
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0000011111 => 0000011111 => 1
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0000011111 => 0000011111 => 1
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0000011111 => 0000011111 => 1
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0000011111 => 0000011111 => 1
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 0000011111 => 0000011111 => 1
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 0000011111 => 0000011111 => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => 000000111111 => 000000111111 => 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => 110011001100 => 000011001111 => 010000101111 => 3
[1,1,0,0,1,1,1,0,0,0,1,0]
 => 110011100010 => 000010011111 => 010000011111 => 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 110011100100 => 000010011111 => 010000011111 => 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 110011110000 => 000000111111 => 000000111111 => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => 110111100000 => 000000111111 => 000000111111 => 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => 111000110010 => 000011001111 => 010000101111 => 3
[1,1,1,0,0,0,1,1,1,0,0,0]
 => 111000111000 => 000000111111 => 000000111111 => 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => 111001001100 => 000010011111 => 010000011111 => 2
[1,1,1,0,0,1,1,0,0,0,1,0]
 => 111001100010 => 000010011111 => 010000011111 => 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => 111001100100 => 000010011111 => 010000011111 => 2
[1,1,1,0,0,1,1,1,0,0,0,0]
 => 111001110000 => 000000111111 => 000000111111 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => 111011100000 => 000000111111 => 000000111111 => 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => 111100001100 => 000000111111 => 000000111111 => 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => 111100010010 => 000010011111 => 010000011111 => 2
Description
The number of runs of ones in a binary word.
Matching statistic: St000291
(load all 2 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 01 => 01 => 0 = 1 - 1
[1,0,1,0]
 => 1010 => 0011 => 0011 => 0 = 1 - 1
[1,1,0,0]
 => 1100 => 0011 => 0011 => 0 = 1 - 1
[1,0,1,0,1,0]
 => 101010 => 001011 => 100011 => 1 = 2 - 1
[1,0,1,1,0,0]
 => 101100 => 000111 => 000111 => 0 = 1 - 1
[1,1,0,0,1,0]
 => 110010 => 000111 => 000111 => 0 = 1 - 1
[1,1,0,1,0,0]
 => 110100 => 000111 => 000111 => 0 = 1 - 1
[1,1,1,0,0,0]
 => 111000 => 000111 => 000111 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 00101011 => 11000011 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 00010111 => 10000111 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 00010111 => 10000111 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 00010111 => 10000111 => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 00001111 => 00001111 => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 00010111 => 10000111 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 00001111 => 00001111 => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 00010111 => 10000111 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 00010111 => 10000111 => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 00001111 => 00001111 => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 00001111 => 00001111 => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 00001111 => 00001111 => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 00001111 => 00001111 => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 00001111 => 00001111 => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0001001111 => 0100001111 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0001001111 => 0100001111 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 0000011111 => 0000011111 => 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => 000000111111 => 000000111111 => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => 110011001100 => 000011001111 => 010000101111 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => 110011100010 => 000010011111 => 010000011111 => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 110011100100 => 000010011111 => 010000011111 => 1 = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 110011110000 => 000000111111 => 000000111111 => 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => 110111100000 => 000000111111 => 000000111111 => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => 111000110010 => 000011001111 => 010000101111 => 2 = 3 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => 111000111000 => 000000111111 => 000000111111 => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => 111001001100 => 000010011111 => 010000011111 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
 => 111001100010 => 000010011111 => 010000011111 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => 111001100100 => 000010011111 => 010000011111 => 1 = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => 111001110000 => 000000111111 => 000000111111 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => 111011100000 => 000000111111 => 000000111111 => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => 111100001100 => 000000111111 => 000000111111 => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => 111100010010 => 000010011111 => 010000011111 => 1 = 2 - 1
Description
The number of descents of a binary word.
Matching statistic: St000807
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000807: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 01 => [1,1] => 0 = 1 - 1
[1,0,1,0]
 => 1010 => 0011 => [2,2] => 0 = 1 - 1
[1,1,0,0]
 => 1100 => 0011 => [2,2] => 0 = 1 - 1
[1,0,1,0,1,0]
 => 101010 => 001011 => [2,1,1,2] => 1 = 2 - 1
[1,0,1,1,0,0]
 => 101100 => 000111 => [3,3] => 0 = 1 - 1
[1,1,0,0,1,0]
 => 110010 => 000111 => [3,3] => 0 = 1 - 1
[1,1,0,1,0,0]
 => 110100 => 000111 => [3,3] => 0 = 1 - 1
[1,1,1,0,0,0]
 => 111000 => 000111 => [3,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 00101011 => [2,1,1,1,1,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 00010111 => [3,1,1,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 00010111 => [3,1,1,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 00010111 => [3,1,1,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 00001111 => [4,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 00010111 => [3,1,1,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 00001111 => [4,4] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 00010111 => [3,1,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 00010111 => [3,1,1,3] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 00001111 => [4,4] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 00001111 => [4,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 00001111 => [4,4] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 00001111 => [4,4] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 00001111 => [4,4] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0001001111 => [3,1,2,4] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0001001111 => [3,1,2,4] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 0000011111 => [5,5] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 0000011111 => [5,5] => 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => 101111100000 => 000000111111 => [6,6] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => 110011001100 => 000011001111 => [4,2,2,4] => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => 110011100010 => 000010011111 => [4,1,2,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => 110011100100 => 000010011111 => [4,1,2,5] => 1 = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => 110011110000 => 000000111111 => [6,6] => 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => 110111100000 => 000000111111 => [6,6] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => 111000110010 => 000011001111 => [4,2,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => 111000111000 => 000000111111 => [6,6] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => 111001001100 => 000010011111 => [4,1,2,5] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
 => 111001100010 => 000010011111 => [4,1,2,5] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => 111001100100 => 000010011111 => [4,1,2,5] => 1 = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => 111001110000 => 000000111111 => [6,6] => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => 111011100000 => 000000111111 => [6,6] => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => 111100001100 => 000000111111 => [6,6] => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => 111100010010 => 000010011111 => [4,1,2,5] => 1 = 2 - 1
Description
The sum of the heights of the valleys of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. A valley is a contiguous subsequence consisting of an up step, a sequence of horizontal steps, and a down step. This statistic is the sum of the heights of the valleys.
Matching statistic: St000871
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000871: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => []
 => []
 => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1]
 => [1,0]
 => [2,1] => 0 = 1 - 1
[1,1,0,0]
 => []
 => []
 => [1] => 0 = 1 - 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1]
 => [1,0]
 => [2,1] => 0 = 1 - 1
[1,1,1,0,0,0]
 => []
 => []
 => [1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0]
 => [2,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [1] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0]
 => [2,1] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [1] => 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [6,8,5,1,2,7,3,4] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [9,1,2,3,6,7,8,4,5] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [8,1,2,5,6,7,3,4] => ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [4,1,2,5,6,7,8,3] => ? = 3 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,1,2,3,6,7,8,4] => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 1 = 2 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 0 = 1 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,1,0,0]
 => [2,3,1] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 0 = 1 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,0,1,0]
 => [3,1,2] => 0 = 1 - 1
Description
The number of very big ascents of a permutation. A very big ascent of a permutation $\pi$ is an index $i$ such that $\pi_{i+1} - \pi_i > 2$. For the number of ascents, see [[St000245]] and for the number of big ascents, see [[St000646]]. General $r$-ascents were for example be studied in [1, Section 2].
Matching statistic: St001673
(load all 3 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 85%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => []
 => []
 => [0] => ? = 1 - 1
[1,0,1,0]
 => [1]
 => [[1]]
 => [1] => 0 = 1 - 1
[1,1,0,0]
 => []
 => []
 => [0] => ? = 1 - 1
[1,0,1,0,1,0]
 => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 0 = 1 - 1
[1,1,0,0,1,0]
 => [2]
 => [[1,2]]
 => [2] => 0 = 1 - 1
[1,1,0,1,0,0]
 => [1]
 => [[1]]
 => [1] => 0 = 1 - 1
[1,1,1,0,0,0]
 => []
 => []
 => [0] => ? = 1 - 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [[1,2,3]]
 => [3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [[1,2]]
 => [2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [[1]]
 => [1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [0] => ? = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [4,2,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[1,2,3,4]]
 => [4] => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[1,2,3]]
 => [3] => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[1,2]]
 => [2] => 0 = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [[1]]
 => [1] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [0] => ? = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => [4,4,2,2] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11]]
 => [5,2,2,2] => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [4,2,2,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [2,2,2,2] => 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => [[1,2,3,4,5],[6,7,8],[9,10,11]]
 => [5,3,3] => ? = 3 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [3,3,3] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2]
 => [[1,2,3,4],[5,6,7,8],[9,10]]
 => [4,4,2] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [5,2,2] => 1 = 2 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [4,2,2] => 1 = 2 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [4,4] => 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [5,3] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => 1 = 2 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 0 = 1 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [1,1] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[1,2,3,4,5]]
 => [5] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[1,2,3,4]]
 => [4] => 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[1,2,3]]
 => [3] => 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => [0] => ? = 1 - 1
Description
The degree of asymmetry of an integer composition. This is the number of pairs of symmetrically positioned distinct entries.
Matching statistic: St000846
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 84%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,3,4,5,6,2] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,4,2,5,6,3] => ([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,4,5,6,2,3] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,5,6,2,3,4] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,2,5,6,3,4] => ([(0,5),(3,2),(4,1),(5,3),(5,4)],6)
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,6,2,3,4,5] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,2,6,3,4,5] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,2,3,6,4,5] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [1,3,4,5,6,7,2] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
 => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [1,4,2,5,6,7,3] => ([(0,3),(0,5),(3,6),(4,2),(5,1),(5,6),(6,4)],7)
 => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [1,4,5,2,6,7,3] => ([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
 => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [1,4,5,6,7,2,3] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [1,2,4,5,6,7,3] => ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
 => 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [1,5,2,6,7,3,4] => ([(0,3),(0,5),(3,6),(4,1),(5,4),(5,6),(6,2)],7)
 => ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [1,5,6,7,2,3,4] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
 => ? = 2
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [1,5,2,3,6,7,4] => ([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7)
 => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [1,2,5,3,6,7,4] => ([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
 => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [1,2,5,6,7,3,4] => ([(0,6),(3,4),(4,1),(5,2),(6,3),(6,5)],7)
 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,2,3,5,6,7,4] => ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
 => 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [1,6,7,2,3,4,5] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [1,6,2,3,7,4,5] => ([(0,2),(0,5),(2,6),(3,1),(4,3),(4,6),(5,4)],7)
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [1,2,6,7,3,4,5] => ([(0,6),(3,4),(4,1),(5,2),(6,3),(6,5)],7)
 => 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [1,6,2,3,4,7,5] => ([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
 => ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [1,2,6,3,4,7,5] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
 => ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [1,2,3,6,7,4,5] => ([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7)
 => 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,2,3,4,6,7,5] => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [1,7,2,3,4,5,6] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [1,2,7,3,4,5,6] => ([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7)
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,2,3,7,4,5,6] => ([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7)
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,2,3,4,7,5,6] => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7)
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
 => 1
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St001503
(load all 49 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001503: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => []
 => []
 => [1,0]
 => 1
[1,0,1,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0]
 => []
 => []
 => [1,0]
 => 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => []
 => []
 => [1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [1,0]
 => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => ? = 2
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => [1,0]
 => 1
Description
The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.
Matching statistic: St001290
(load all 2 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001290: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 79%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => []
 => []
 => [1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0]
 => []
 => []
 => [1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => []
 => []
 => [1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [1,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [1,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? = 3 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => ? = 2 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => ? = 3 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 2 + 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => [1,0]
 => 2 = 1 + 1
Description
The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A.
Matching statistic: St000402
(load all 2 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000402: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 77%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => []
 => []
 => [] => ? = 1
[1,0,1,0]
 => [1]
 => [[1]]
 => [1] => ? = 1
[1,1,0,0]
 => []
 => []
 => [] => ? = 1
[1,0,1,0,1,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[1,0,1,1,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,1,0,0,1,0]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[1,1,0,1,0,0]
 => [1]
 => [[1]]
 => [1] => ? = 1
[1,1,1,0,0,0]
 => []
 => []
 => [] => ? = 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => 2
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 2
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 2
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => 2
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => 2
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [[1,2],[3]]
 => [3,1,2] => 2
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => [[1]]
 => [1] => ? = 1
[1,1,1,1,0,0,0,0]
 => []
 => []
 => [] => ? = 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => 2
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [[1]]
 => [1] => ? = 1
[1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => [] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => [11,12,9,10,5,6,7,8,1,2,3,4] => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2]
 => [[1,2,3,4,5],[6,7],[8,9],[10,11]]
 => [10,11,8,9,6,7,1,2,3,4,5] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => [9,10,7,8,5,6,1,2,3,4] => 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [5,3,3]
 => [[1,2,3,4,5],[6,7,8],[9,10,11]]
 => [9,10,11,6,7,8,1,2,3,4,5] => ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => [7,8,9,4,5,6,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,4,2]
 => [[1,2,3,4],[5,6,7,8],[9,10]]
 => [9,10,5,6,7,8,1,2,3,4] => 2
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [5,2,2]
 => [[1,2,3,4,5],[6,7],[8,9]]
 => [8,9,6,7,1,2,3,4,5] => 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => [7,8,5,6,1,2,3,4] => 2
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [6,7,8,1,2,3,4,5] => 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [[1,2]]
 => [1,2] => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [[1]]
 => [1] => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => [] => ? = 1
Description
Half the size of the symmetry class of a permutation. The symmetry class of a permutation $\pi$ is the set of all permutations that can be obtained from $\pi$ by the three elementary operations '''inverse''' ([[Mp00066]]), '''reverse''' ([[Mp00064]]), and '''complement''' ([[Mp00069]]). This statistic is undefined for the unique permutation on one element, because its value would be $1/2$.
Matching statistic: St000834
(load all 2 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 74%●distinct values known / distinct values provided: 67%
Values
[1,0]
 => [1,1,0,0]
 => [2,1] => [1,2] => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,4,1,2] => 2
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1,2,4] => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,3,1,2] => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,2,3] => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [3,4,5,1,2] => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [3,4,1,2,5] => 2
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [3,5,4,1,2] => 2
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [3,5,1,2,4] => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,1,2,4,5] => 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [4,3,5,1,2] => 2
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [4,3,1,2,5] => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => [4,5,3,1,2] => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => [4,5,1,2,3] => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => [4,1,2,3,5] => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [5,4,3,1,2] => 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [5,4,1,2,3] => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [5,1,2,3,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,5,4,3,1] => [3,1,2,4,5,6] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [3,2,5,4,6,1] => [4,3,6,5,1,2] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,2,6,5,4,1] => [4,3,1,2,5,6] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,6,5,4,2,1] => [4,1,2,3,5,6] => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,3,2,6,5,1] => [5,4,3,1,2,6] => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [4,3,5,2,6,1] => [5,4,6,3,1,2] => 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,3,6,5,2,1] => [5,4,1,2,3,6] => 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [4,6,5,3,2,1] => [5,1,2,3,4,6] => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [5,4,3,2,6,1] => [6,5,4,3,1,2] => 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,4,3,6,2,1] => [6,5,4,1,2,3] => 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [5,4,6,3,2,1] => [6,5,1,2,3,4] => 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [5,6,4,3,2,1] => [6,1,2,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,7,6,5,4,3,1] => [3,1,2,4,5,6,7] => 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,7,6,1] => [4,3,6,5,1,2,7] => ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
 => [3,2,6,5,4,7,1] => [4,3,7,6,5,1,2] => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
 => [3,2,6,5,7,4,1] => [4,3,7,6,1,2,5] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,2,7,6,5,4,1] => [4,3,1,2,5,6,7] => ? = 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,7,6,5,4,2,1] => [4,1,2,3,5,6,7] => 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
 => [4,3,2,6,5,7,1] => [5,4,3,7,6,1,2] => ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => [4,3,2,7,6,5,1] => [5,4,3,1,2,6,7] => ? = 1
[1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [4,3,5,2,7,6,1] => [5,4,6,3,1,2,7] => ? = 2
[1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => [4,3,6,5,2,7,1] => [5,4,7,6,3,1,2] => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [4,3,6,5,7,2,1] => [5,4,7,6,1,2,3] => ? = 2
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [4,3,7,6,5,2,1] => [5,4,1,2,3,6,7] => ? = 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [4,7,6,5,3,2,1] => [5,1,2,3,4,6,7] => 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => [5,4,3,2,7,6,1] => [6,5,4,3,1,2,7] => ? = 1
[1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [5,4,3,6,2,7,1] => [6,5,4,7,3,1,2] => ? = 2
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => [5,4,3,7,6,2,1] => [6,5,4,1,2,3,7] => ? = 1
[1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [5,4,6,3,2,7,1] => [6,5,7,4,3,1,2] => ? = 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => [6,5,7,4,1,2,3] => ? = 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [5,4,7,6,3,2,1] => [6,5,1,2,3,4,7] => ? = 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [5,7,6,4,3,2,1] => [6,1,2,3,4,5,7] => 1
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => [7,6,5,4,3,1,2] => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => [7,6,5,4,1,2,3] => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => [7,6,5,1,2,3,4] => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => [7,6,1,2,3,4,5] => 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => [7,1,2,3,4,5,6] => 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 1
Description
The number of right outer peaks of a permutation. A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$. In other words, it is a peak in the word $[w_1,..., w_n,0]$.
The following 197 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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 $eA$. St001394The genus of a permutation. St000035The number of left outer peaks of a permutation. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000779The tier of a permutation. St000451The length of the longest pattern of the form k 1 2. St000732The number of double deficiencies of a permutation. St001513The number of nested exceedences of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001220The width of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000021The number of descents of a permutation. St000099The number of valleys of a permutation, including the boundary. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001487The number of inner corners of a skew partition. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001874Lusztig's a-function for the symmetric group. St000023The number of inner peaks of a permutation. St000039The number of crossings of a permutation. St000325The width of the tree associated to a permutation. St001470The cyclic holeyness of a permutation. St001731The factorization defect of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001871The number of triconnected components of a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000691The number of changes of a binary word. St000864The number of circled entries of the shifted recording tableau of a permutation. St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000264The girth of a graph, which is not a tree. St001964The interval resolution global dimension of a poset. St000758The length of the longest staircase fitting into an integer composition. St000805The number of peaks of the associated bargraph. St000903The number of different parts of an integer composition. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001884The number of borders of a binary word. St000682The Grundy value of Welter's game on a binary word. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000761The number of ascents in an integer composition. St000767The number of runs in an integer composition. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000253The crossing number of a set partition. St000237The number of small exceedances. St001083The number of boxed occurrences of 132 in a permutation. St001115The number of even descents of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000842The breadth of a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St001862The number of crossings of a signed permutation. St000260The radius of a connected graph. St000405The number of occurrences of the pattern 1324 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001625The Möbius invariant of a lattice. St000455The second largest eigenvalue of a graph if it is integral. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000618The number of self-evacuating tableaux of given shape. St000731The number of double exceedences of a permutation. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001866The nesting alignments of a signed permutation. 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. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001344The neighbouring number of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001857The number of edges in the reduced word graph of a signed permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000255The number of reduced Kogan faces with the permutation as type. St000570The Edelman-Greene number of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001060The distinguishing index of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001413Half the length of the longest even length palindromic prefix of a binary word. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001768The number of reduced words of a signed permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000017The number of inversions of a standard tableau. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000236The number of cyclical small weak excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000308The height of the tree associated to a permutation. St000338The number of pixed points of a permutation. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000370The genus of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000504The cardinality of the first block of a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000636The hull number of a graph. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001062The maximal size of a block of a set partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001309The number of four-cliques in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001411The number of patterns 321 or 3412 in a permutation. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001867The number of alignments of type EN of a signed permutation. St001948The number of augmented double ascents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001624The breadth of a lattice. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph.