- FindStat

Your data matches 74 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000010

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 =>  => [1] => [1]
 => 1
[[1,2]]
 => 0 => [2] => [2]
 => 1
[[1],[2]]
 => 1 => [1,1] => [1,1]
 => 2
[[1,2,3]]
 => 00 => [3] => [3]
 => 1
[[1,3],[2]]
 => 10 => [1,2] => [2,1]
 => 2
[[1,2],[3]]
 => 01 => [2,1] => [2,1]
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[[1,2,3,4]]
 => 000 => [4] => [4]
 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => [3,1]
 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => [2,2]
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => [3,1]
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => [2,1,1]
 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => [2,2]
 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => [2,1,1]
 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => [2,1,1]
 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => [2,1,1]
 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => [5]
 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => [4,1]
 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => [3,2]
 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => [3,2]
 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => [4,1]
 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => [2,2,1]
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => [3,2]
 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => [3,1,1]
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => [2,2,1]
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => [3,2]
 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => [3,1,1]
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => [2,2,1]
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => [2,2,1]
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => [3,1,1]
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => [2,2,1]
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => [3,1,1]
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => [2,1,1,1]
 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => [2,2,1]
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => [2,2,1]
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => [2,1,1,1]
 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => [2,2,1]
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => [2,1,1,1]
 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => [2,1,1,1]
 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => [2,1,1,1]
 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => [2,1,1,1]
 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => [6]
 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => [5,1]
 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => [4,2]
 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => [3,3]
 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => [4,2]
 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => [5,1]
 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => [3,2,1]
 => 3

Description

The length of the partition.

Matching statistic: St000288
(load all 52 compositions to match this statistic)

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 92%

Values

[[1]]
 =>  =>  =>  => ? = 1 - 1
[[1,2]]
 => 0 => 0 => 0 => 0 = 1 - 1
[[1],[2]]
 => 1 => 1 => 1 => 1 = 2 - 1
[[1,2,3]]
 => 00 => 00 => 00 => 0 = 1 - 1
[[1,3],[2]]
 => 10 => 01 => 01 => 1 = 2 - 1
[[1,2],[3]]
 => 01 => 10 => 01 => 1 = 2 - 1
[[1],[2],[3]]
 => 11 => 11 => 11 => 2 = 3 - 1
[[1,2,3,4]]
 => 000 => 000 => 000 => 0 = 1 - 1
[[1,3,4],[2]]
 => 100 => 001 => 001 => 1 = 2 - 1
[[1,2,4],[3]]
 => 010 => 010 => 001 => 1 = 2 - 1
[[1,2,3],[4]]
 => 001 => 100 => 001 => 1 = 2 - 1
[[1,3],[2,4]]
 => 101 => 101 => 011 => 2 = 3 - 1
[[1,2],[3,4]]
 => 010 => 010 => 001 => 1 = 2 - 1
[[1,4],[2],[3]]
 => 110 => 011 => 011 => 2 = 3 - 1
[[1,3],[2],[4]]
 => 101 => 101 => 011 => 2 = 3 - 1
[[1,2],[3],[4]]
 => 011 => 110 => 011 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => 111 => 111 => 111 => 3 = 4 - 1
[[1,2,3,4,5]]
 => 0000 => 0000 => 0000 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => 1000 => 0001 => 0001 => 1 = 2 - 1
[[1,2,4,5],[3]]
 => 0100 => 0010 => 0001 => 1 = 2 - 1
[[1,2,3,5],[4]]
 => 0010 => 0100 => 0001 => 1 = 2 - 1
[[1,2,3,4],[5]]
 => 0001 => 1000 => 0001 => 1 = 2 - 1
[[1,3,5],[2,4]]
 => 1010 => 0101 => 0101 => 2 = 3 - 1
[[1,2,5],[3,4]]
 => 0100 => 0010 => 0001 => 1 = 2 - 1
[[1,3,4],[2,5]]
 => 1001 => 1001 => 0011 => 2 = 3 - 1
[[1,2,4],[3,5]]
 => 0101 => 1010 => 0011 => 2 = 3 - 1
[[1,2,3],[4,5]]
 => 0010 => 0100 => 0001 => 1 = 2 - 1
[[1,4,5],[2],[3]]
 => 1100 => 0011 => 0011 => 2 = 3 - 1
[[1,3,5],[2],[4]]
 => 1010 => 0101 => 0101 => 2 = 3 - 1
[[1,2,5],[3],[4]]
 => 0110 => 0110 => 0011 => 2 = 3 - 1
[[1,3,4],[2],[5]]
 => 1001 => 1001 => 0011 => 2 = 3 - 1
[[1,2,4],[3],[5]]
 => 0101 => 1010 => 0011 => 2 = 3 - 1
[[1,2,3],[4],[5]]
 => 0011 => 1100 => 0011 => 2 = 3 - 1
[[1,4],[2,5],[3]]
 => 1101 => 1011 => 0111 => 3 = 4 - 1
[[1,3],[2,5],[4]]
 => 1010 => 0101 => 0101 => 2 = 3 - 1
[[1,2],[3,5],[4]]
 => 0110 => 0110 => 0011 => 2 = 3 - 1
[[1,3],[2,4],[5]]
 => 1011 => 1101 => 0111 => 3 = 4 - 1
[[1,2],[3,4],[5]]
 => 0101 => 1010 => 0011 => 2 = 3 - 1
[[1,5],[2],[3],[4]]
 => 1110 => 0111 => 0111 => 3 = 4 - 1
[[1,4],[2],[3],[5]]
 => 1101 => 1011 => 0111 => 3 = 4 - 1
[[1,3],[2],[4],[5]]
 => 1011 => 1101 => 0111 => 3 = 4 - 1
[[1,2],[3],[4],[5]]
 => 0111 => 1110 => 0111 => 3 = 4 - 1
[[1],[2],[3],[4],[5]]
 => 1111 => 1111 => 1111 => 4 = 5 - 1
[[1,2,3,4,5,6]]
 => 00000 => 00000 => 00000 => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => 10000 => 00001 => 00001 => 1 = 2 - 1
[[1,2,4,5,6],[3]]
 => 01000 => 00010 => 00001 => 1 = 2 - 1
[[1,2,3,5,6],[4]]
 => 00100 => 00100 => 00001 => 1 = 2 - 1
[[1,2,3,4,6],[5]]
 => 00010 => 01000 => 00001 => 1 = 2 - 1
[[1,2,3,4,5],[6]]
 => 00001 => 10000 => 00001 => 1 = 2 - 1
[[1,3,5,6],[2,4]]
 => 10100 => 00101 => 00101 => 2 = 3 - 1
[[1,2,5,6],[3,4]]
 => 01000 => 00010 => 00001 => 1 = 2 - 1
[[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => 10101010101 => 10101010101 => 01010101011 => ? = 7 - 1
[[1,3,5,7,9,10],[2,4,6,8,11,12]]
 => 10101010010 => 01001010101 => 00101010101 => ? = 6 - 1
[[1,3,5,7,8,11],[2,4,6,9,10,12]]
 => 10101001001 => 10010010101 => 00100101011 => ? = 6 - 1
[[1,3,5,7,8,10],[2,4,6,9,11,12]]
 => 10101001010 => 01010010101 => 00101010101 => ? = 6 - 1
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
 => 10101000100 => 00100010101 => 00010010101 => ? = 5 - 1
[[1,3,5,6,9,11],[2,4,7,8,10,12]]
 => 10100100101 => 10100100101 => 00100101011 => ? = 6 - 1
[[1,3,5,6,9,10],[2,4,7,8,11,12]]
 => 10100100010 => 01000100101 => 00010010101 => ? = 5 - 1
[[1,3,5,6,8,11],[2,4,7,9,10,12]]
 => 10100101001 => 10010100101 => 00100101011 => ? = 6 - 1
[[1,3,5,6,8,10],[2,4,7,9,11,12]]
 => 10100101010 => 01010100101 => 00101010101 => ? = 6 - 1
[[1,3,5,6,7,11],[2,4,8,9,10,12]]
 => 10100010001 => 10001000101 => 00010001011 => ? = 5 - 1
[[1,3,5,6,7,10],[2,4,8,9,11,12]]
 => 10100010010 => 01001000101 => 00010010101 => ? = 5 - 1
[[1,3,5,6,7,9],[2,4,8,10,11,12]]
 => 10100010100 => 00101000101 => 00010010101 => ? = 5 - 1
[[1,3,5,6,7,8],[2,4,9,10,11,12]]
 => 10100001000 => 00010000101 => 00001000101 => ? = 4 - 1
[[1,3,4,7,9,11],[2,5,6,8,10,12]]
 => 10010010101 => 10101001001 => 00100101011 => ? = 6 - 1
[[1,3,4,7,8,11],[2,5,6,9,10,12]]
 => 10010001001 => 10010001001 => 00010010011 => ? = 5 - 1
[[1,3,4,7,8,10],[2,5,6,9,11,12]]
 => 10010001010 => 01010001001 => 00010010101 => ? = 5 - 1
[[1,3,4,7,8,9],[2,5,6,10,11,12]]
 => 10010000100 => 00100001001 => 00001001001 => ? = 4 - 1
[[1,3,4,6,9,11],[2,5,7,8,10,12]]
 => 10010100101 => 10100101001 => 00100101011 => ? = 6 - 1
[[1,3,4,6,9,10],[2,5,7,8,11,12]]
 => 10010100010 => 01000101001 => 00010010101 => ? = 5 - 1
[[1,3,4,6,8,11],[2,5,7,9,10,12]]
 => 10010101001 => 10010101001 => 00100101011 => ? = 6 - 1
[[1,3,4,6,8,10],[2,5,7,9,11,12]]
 => 10010101010 => 01010101001 => 00101010101 => ? = 6 - 1
[[1,3,4,6,7,11],[2,5,8,9,10,12]]
 => 10010010001 => 10001001001 => 00010010011 => ? = 5 - 1
[[1,3,4,5,9,11],[2,6,7,8,10,12]]
 => 10001000101 => 10100010001 => 00010001011 => ? = 5 - 1
[[1,3,4,5,9,10],[2,6,7,8,11,12]]
 => 10001000010 => 01000010001 => 00001000101 => ? = 4 - 1
[[1,3,4,5,8,11],[2,6,7,9,10,12]]
 => 10001001001 => 10010010001 => 00010010011 => ? = 5 - 1
[[1,3,4,5,8,10],[2,6,7,9,11,12]]
 => 10001001010 => 01010010001 => 00010010101 => ? = 5 - 1
[[1,3,4,5,7,11],[2,6,8,9,10,12]]
 => 10001010001 => 10001010001 => 00010001011 => ? = 5 - 1
[[1,3,4,5,7,10],[2,6,8,9,11,12]]
 => 10001010010 => 01001010001 => 00010010101 => ? = 5 - 1
[[1,3,4,5,7,9],[2,6,8,10,11,12]]
 => 10001010100 => 00101010001 => 00010010101 => ? = 5 - 1
[[1,3,4,5,6,11],[2,7,8,9,10,12]]
 => 10000100001 => 10000100001 => 00001000011 => ? = 4 - 1
[[1,3,4,5,6,10],[2,7,8,9,11,12]]
 => 10000100010 => 01000100001 => 00001000101 => ? = 4 - 1
[[1,3,4,5,6,9],[2,7,8,10,11,12]]
 => 10000100100 => 00100100001 => 00001001001 => ? = 4 - 1
[[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => 10000101000 => 00010100001 => 00001000101 => ? = 4 - 1
[[1,3,4,5,6,7],[2,8,9,10,11,12]]
 => 10000010000 => 00001000001 => 00000100001 => ? = 3 - 1
[[1,2,5,7,9,11],[3,4,6,8,10,12]]
 => 01001010101 => 10101010010 => 00010101011 => ? = 6 - 1
[[1,2,5,7,9,10],[3,4,6,8,11,12]]
 => 01001010010 => 01001010010 => 00010010101 => ? = 5 - 1
[[1,2,5,7,8,11],[3,4,6,9,10,12]]
 => 01001001001 => 10010010010 => 00010010011 => ? = 5 - 1
[[1,2,5,7,8,10],[3,4,6,9,11,12]]
 => 01001001010 => 01010010010 => 00010010101 => ? = 5 - 1
[[1,2,5,7,8,9],[3,4,6,10,11,12]]
 => 01001000100 => 00100010010 => 00001001001 => ? = 4 - 1
[[1,2,5,6,9,11],[3,4,7,8,10,12]]
 => 01000100101 => 10100100010 => 00001001011 => ? = 5 - 1
[[1,2,5,6,9,10],[3,4,7,8,11,12]]
 => 01000100010 => 01000100010 => 00001000101 => ? = 4 - 1
[[1,2,5,6,8,11],[3,4,7,9,10,12]]
 => 01000101001 => 10010100010 => 00001001011 => ? = 5 - 1
[[1,2,5,6,8,10],[3,4,7,9,11,12]]
 => 01000101010 => 01010100010 => 00001010101 => ? = 5 - 1
[[1,2,5,6,8,9],[3,4,7,10,11,12]]
 => 01000100100 => 00100100010 => 00001001001 => ? = 4 - 1
[[1,2,5,6,7,11],[3,4,8,9,10,12]]
 => 01000010001 => 10001000010 => 00000100011 => ? = 4 - 1
[[1,2,5,6,7,10],[3,4,8,9,11,12]]
 => 01000010010 => 01001000010 => 00000100101 => ? = 4 - 1
[[1,2,5,6,7,9],[3,4,8,10,11,12]]
 => 01000010100 => 00101000010 => 00000100101 => ? = 4 - 1
[[1,2,5,6,7,8],[3,4,9,10,11,12]]
 => 01000001000 => 00010000010 => 00000010001 => ? = 3 - 1
[[1,2,4,7,9,11],[3,5,6,8,10,12]]
 => 01010010101 => 10101001010 => 00010101011 => ? = 6 - 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000157
(load all 4 compositions to match this statistic)

Mapping

St000157: Standard tableaux ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 92%

Values

[[1]]
 => 0 = 1 - 1
[[1,2]]
 => 0 = 1 - 1
[[1],[2]]
 => 1 = 2 - 1
[[1,2,3]]
 => 0 = 1 - 1
[[1,3],[2]]
 => 1 = 2 - 1
[[1,2],[3]]
 => 1 = 2 - 1
[[1],[2],[3]]
 => 2 = 3 - 1
[[1,2,3,4]]
 => 0 = 1 - 1
[[1,3,4],[2]]
 => 1 = 2 - 1
[[1,2,4],[3]]
 => 1 = 2 - 1
[[1,2,3],[4]]
 => 1 = 2 - 1
[[1,3],[2,4]]
 => 2 = 3 - 1
[[1,2],[3,4]]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => 2 = 3 - 1
[[1,3],[2],[4]]
 => 2 = 3 - 1
[[1,2],[3],[4]]
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => 3 = 4 - 1
[[1,2,3,4,5]]
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => 1 = 2 - 1
[[1,2,4,5],[3]]
 => 1 = 2 - 1
[[1,2,3,5],[4]]
 => 1 = 2 - 1
[[1,2,3,4],[5]]
 => 1 = 2 - 1
[[1,3,5],[2,4]]
 => 2 = 3 - 1
[[1,2,5],[3,4]]
 => 1 = 2 - 1
[[1,3,4],[2,5]]
 => 2 = 3 - 1
[[1,2,4],[3,5]]
 => 2 = 3 - 1
[[1,2,3],[4,5]]
 => 1 = 2 - 1
[[1,4,5],[2],[3]]
 => 2 = 3 - 1
[[1,3,5],[2],[4]]
 => 2 = 3 - 1
[[1,2,5],[3],[4]]
 => 2 = 3 - 1
[[1,3,4],[2],[5]]
 => 2 = 3 - 1
[[1,2,4],[3],[5]]
 => 2 = 3 - 1
[[1,2,3],[4],[5]]
 => 2 = 3 - 1
[[1,4],[2,5],[3]]
 => 3 = 4 - 1
[[1,3],[2,5],[4]]
 => 2 = 3 - 1
[[1,2],[3,5],[4]]
 => 2 = 3 - 1
[[1,3],[2,4],[5]]
 => 3 = 4 - 1
[[1,2],[3,4],[5]]
 => 2 = 3 - 1
[[1,5],[2],[3],[4]]
 => 3 = 4 - 1
[[1,4],[2],[3],[5]]
 => 3 = 4 - 1
[[1,3],[2],[4],[5]]
 => 3 = 4 - 1
[[1,2],[3],[4],[5]]
 => 3 = 4 - 1
[[1],[2],[3],[4],[5]]
 => 4 = 5 - 1
[[1,2,3,4,5,6]]
 => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => 1 = 2 - 1
[[1,2,4,5,6],[3]]
 => 1 = 2 - 1
[[1,2,3,5,6],[4]]
 => 1 = 2 - 1
[[1,2,3,4,6],[5]]
 => 1 = 2 - 1
[[1,2,3,4,5],[6]]
 => 1 = 2 - 1
[[1,3,5,6],[2,4]]
 => 2 = 3 - 1
[[1,2,3,4],[5,6],[7,8],[9,10],[11,12]]
 => ? = 5 - 1
[[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 12 - 1
[[1,2,3,4,5,6,7,9],[8]]
 => ? = 2 - 1
[[1,2,3,4,5,6,9],[7,8]]
 => ? = 2 - 1
[[1,2,3,4,5,6,9],[7],[8]]
 => ? = 3 - 1
[[1,2,3,4,5,9],[6,7,8]]
 => ? = 2 - 1
[[1,2,3,4,5,9],[6,7],[8]]
 => ? = 3 - 1
[[1,2,3,4,5,9],[6],[7],[8]]
 => ? = 4 - 1
[[1,2,3,4,9],[5,6,7],[8]]
 => ? = 3 - 1
[[1,2,3,4,9],[5,6],[7],[8]]
 => ? = 4 - 1
[[1,2,3,4,9],[5],[6],[7],[8]]
 => ? = 5 - 1
[[1,2,3,7],[4,5,6,9],[8]]
 => ? = 3 - 1
[[1,2,3,9],[4,5,6],[7,8]]
 => ? = 3 - 1
[[1,2,3,9],[4,5,6],[7],[8]]
 => ? = 4 - 1
[[1,2,3,9],[4,5],[6,7],[8]]
 => ? = 4 - 1
[[1,2,3,9],[4,5],[6],[7],[8]]
 => ? = 5 - 1
[[1,2,3,9],[4],[5],[6],[7],[8]]
 => ? = 6 - 1
[[1,2,5],[3,4,8],[6,7,9]]
 => ? = 4 - 1
[[1,2,5],[3,4,9],[6,7],[8]]
 => ? = 4 - 1
[[1,2,5],[3,4,9],[6],[7],[8]]
 => ? = 5 - 1
[[1,2,9],[3,4],[5,6],[7],[8]]
 => ? = 5 - 1
[[1,2,9],[3,4],[5],[6],[7],[8]]
 => ? = 6 - 1
[[1,3],[2,5],[4,9],[6],[7],[8]]
 => ? = 6 - 1
[[1,3],[2,9],[4],[5],[6],[7],[8]]
 => ? = 7 - 1
[[1,2,3,4,5,6,7,10],[8,9]]
 => ? = 2 - 1
[[1,2,3,4,5,6,7,10],[8],[9]]
 => ? = 3 - 1
[[1,2,3,4,5,6,10],[7,8],[9]]
 => ? = 3 - 1
[[1,2,3,4,5,6,10],[7],[8],[9]]
 => ? = 4 - 1
[[1,2,3,4,5,10],[6,7,8,9]]
 => ? = 2 - 1
[[1,2,3,4,5,10],[6,7,8],[9]]
 => ? = 3 - 1
[[1,2,3,4,5,10],[6,7],[8,9]]
 => ? = 3 - 1
[[1,2,3,4,5,10],[6,7],[8],[9]]
 => ? = 4 - 1
[[1,2,3,4,5,10],[6],[7],[8],[9]]
 => ? = 5 - 1
[[1,2,3,4,10],[5,6,7,8],[9]]
 => ? = 3 - 1
[[1,2,3,4,10],[5,6,7],[8,9]]
 => ? = 3 - 1
[[1,2,3,4,10],[5,6,7],[8],[9]]
 => ? = 4 - 1
[[1,2,3,4,10],[5,6],[7,8],[9]]
 => ? = 4 - 1
[[1,2,3,4,10],[5,6],[7],[8],[9]]
 => ? = 5 - 1
[[1,2,3,4,10],[5],[6],[7],[8],[9]]
 => ? = 6 - 1
[[1,2,3,7],[4,5,6,10],[8,9]]
 => ? = 3 - 1
[[1,2,3,7],[4,5,6,10],[8],[9]]
 => ? = 4 - 1
[[1,2,3,10],[4,5,6],[7,8],[9]]
 => ? = 4 - 1
[[1,2,3,10],[4,5,6],[7],[8],[9]]
 => ? = 5 - 1
[[1,2,3,10],[4,5],[6,7],[8,9]]
 => ? = 4 - 1
[[1,2,3,10],[4,5],[6,7],[8],[9]]
 => ? = 5 - 1
[[1,2,3,10],[4,5],[6],[7],[8],[9]]
 => ? = 6 - 1
[[1,2,3,10],[4],[5],[6],[7],[8],[9]]
 => ? = 7 - 1
[[1,2,5],[3,4,8],[6,7,10],[9]]
 => ? = 4 - 1
[[1,2,5],[3,4,10],[6,7],[8,9]]
 => ? = 4 - 1
[[1,2,5],[3,4,10],[6,7],[8],[9]]
 => ? = 5 - 1

Description

The number of descents of a standard tableau. Entry

$i$ of a standard Young tableau is a descent if

$i+1$ appears in a row below the row of

$i$ .

Matching statistic: St000097

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 83%

Values

[[1]]
 =>  => [1] => ([],1)
 => 1
[[1,2]]
 => 0 => [2] => ([],2)
 => 1
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[[1,2,3]]
 => 00 => [3] => ([],3)
 => 1
[[1,3],[2]]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,3,4]]
 => 000 => [4] => ([],4)
 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => ([],5)
 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => ([(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => ([],6)
 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => ([(4,5)],6)
 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3,5,7,9],[2,4,6,8,10]]
 => 101010101 => [1,2,2,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 6
[[1,3,5,7,8],[2,4,6,9,10]]
 => 101010010 => [1,2,2,3,2] => ([(1,9),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,3,5,6,9],[2,4,7,8,10]]
 => 101001001 => [1,2,3,3,1] => ([(0,9),(1,9),(2,8),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,3,5,6,8],[2,4,7,9,10]]
 => 101001010 => [1,2,3,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,3,5,6,7],[2,4,8,9,10]]
 => 101000100 => [1,2,4,3] => ([(2,9),(3,9),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,3,4,7,9],[2,5,6,8,10]]
 => 100100101 => [1,3,3,2,1] => ([(0,9),(1,8),(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,3,4,7,8],[2,5,6,9,10]]
 => 100100010 => [1,3,4,2] => ([(1,9),(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,3,4,6,9],[2,5,7,8,10]]
 => 100101001 => [1,3,2,3,1] => ([(0,9),(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,3,4,6,8],[2,5,7,9,10]]
 => 100101010 => [1,3,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,3,4,6,7],[2,5,8,9,10]]
 => 100100100 => [1,3,3,3] => ([(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,3,4,5,9],[2,6,7,8,10]]
 => 100010001 => [1,4,4,1] => ([(0,9),(1,9),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,3,4,5,8],[2,6,7,9,10]]
 => 100010010 => [1,4,3,2] => ([(1,9),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,3,4,5,7],[2,6,8,9,10]]
 => 100010100 => [1,4,2,3] => ([(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,3,4,5,6],[2,7,8,9,10]]
 => 100001000 => [1,5,4] => ([(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,5,7,9],[3,4,6,8,10]]
 => 010010101 => [2,3,2,2,1] => ([(0,9),(1,8),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,2,5,7,8],[3,4,6,9,10]]
 => 010010010 => [2,3,3,2] => ([(1,9),(2,9),(3,8),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,5,6,9],[3,4,7,8,10]]
 => 010001001 => [2,4,3,1] => ([(0,9),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,5,6,8],[3,4,7,9,10]]
 => 010001010 => [2,4,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,5,6,7],[3,4,8,9,10]]
 => 010000100 => [2,5,3] => ([(2,9),(3,9),(4,9),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,4,7,9],[3,5,6,8,10]]
 => 010100101 => [2,2,3,2,1] => ([(0,9),(1,8),(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,2,4,7,8],[3,5,6,9,10]]
 => 010100010 => [2,2,4,2] => ([(1,9),(2,9),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,4,6,9],[3,5,7,8,10]]
 => 010101001 => [2,2,2,3,1] => ([(0,9),(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,2,4,6,8],[3,5,7,9,10]]
 => 010101010 => [2,2,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 5
[[1,2,4,6,7],[3,5,8,9,10]]
 => 010100100 => [2,2,3,3] => ([(2,9),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,4,5,9],[3,6,7,8,10]]
 => 010010001 => [2,3,4,1] => ([(0,9),(1,9),(2,9),(3,8),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,4,5,8],[3,6,7,9,10]]
 => 010010010 => [2,3,3,2] => ([(1,9),(2,9),(3,8),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,4,5,7],[3,6,8,9,10]]
 => 010010100 => [2,3,2,3] => ([(2,9),(3,8),(3,9),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,4,5,6],[3,7,8,9,10]]
 => 010001000 => [2,4,4] => ([(3,9),(4,9),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,3,7,9],[4,5,6,8,10]]
 => 001000101 => [3,4,2,1] => ([(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,3,7,8],[4,5,6,9,10]]
 => 001000010 => [3,5,2] => ([(1,9),(2,9),(3,9),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,3,6,9],[4,5,7,8,10]]
 => 001001001 => [3,3,3,1] => ([(0,9),(1,9),(2,8),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,3,6,8],[4,5,7,9,10]]
 => 001001010 => [3,3,2,2] => ([(1,9),(2,8),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,3,6,7],[4,5,8,9,10]]
 => 001000100 => [3,4,3] => ([(2,9),(3,9),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,3,5,9],[4,6,7,8,10]]
 => 001010001 => [3,2,4,1] => ([(0,9),(1,9),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,3,5,8],[4,6,7,9,10]]
 => 001010010 => [3,2,3,2] => ([(1,9),(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,3,5,7],[4,6,8,9,10]]
 => 001010100 => [3,2,2,3] => ([(2,9),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4
[[1,2,3,5,6],[4,7,8,9,10]]
 => 001001000 => [3,3,4] => ([(3,9),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,3,4,9],[5,6,7,8,10]]
 => 000100001 => [4,5,1] => ([(0,9),(1,9),(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,3,4,8],[5,6,7,9,10]]
 => 000100010 => [4,4,2] => ([(1,9),(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,3,4,7],[5,6,8,9,10]]
 => 000100100 => [4,3,3] => ([(2,9),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,3,4,6],[5,7,8,9,10]]
 => 000101000 => [4,2,4] => ([(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
[[1,2,3,4,5],[6,7,8,9,10]]
 => 000010000 => [5,5] => ([(4,9),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 2
[[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => 10101010101 => [1,2,2,2,2,2,1] => ([(0,11),(1,10),(1,11),(2,9),(2,10),(2,11),(3,8),(3,9),(3,10),(3,11),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 7
[[1,3,5,7,9,10],[2,4,6,8,11,12]]
 => 10101010010 => [1,2,2,2,3,2] => ([(1,11),(2,11),(3,10),(3,11),(4,9),(4,10),(4,11),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 6
[[1,3,5,7,8,11],[2,4,6,9,10,12]]
 => 10101001001 => [1,2,2,3,3,1] => ([(0,11),(1,11),(2,10),(2,11),(3,10),(3,11),(4,9),(4,10),(4,11),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 6
[[1,3,5,7,8,10],[2,4,6,9,11,12]]
 => 10101001010 => [1,2,2,3,2,2] => ([(1,11),(2,10),(2,11),(3,10),(3,11),(4,9),(4,10),(4,11),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 6
[[1,3,5,7,8,9],[2,4,6,10,11,12]]
 => 10101000100 => [1,2,2,4,3] => ([(2,11),(3,11),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 5
[[1,3,5,6,9,11],[2,4,7,8,10,12]]
 => 10100100101 => [1,2,3,3,2,1] => ([(0,11),(1,10),(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,9),(4,10),(4,11),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 6
[[1,3,5,6,9,10],[2,4,7,8,11,12]]
 => 10100100010 => [1,2,3,4,2] => ([(1,11),(2,11),(3,11),(4,10),(4,11),(5,10),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 5
[[1,3,5,6,8,11],[2,4,7,9,10,12]]
 => 10100101001 => [1,2,3,2,3,1] => ([(0,11),(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,9),(4,10),(4,11),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
 => ? = 6

Description

The order of the largest clique of the graph. A clique in a graph

$G$ is a subset

$U \subseteq V(G)$ such that any pair of vertices in

$U$ are adjacent. I.e. the subgraph induced by

$U$ is a complete graph.

Matching statistic: St001581

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 58%

Values

[[1]]
 =>  => [1] => ([],1)
 => 1
[[1,2]]
 => 0 => [2] => ([],2)
 => 1
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[[1,2,3]]
 => 00 => [3] => ([],3)
 => 1
[[1,3],[2]]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,3,4]]
 => 000 => [4] => ([],4)
 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => ([],5)
 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => ([(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => ([],6)
 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => ([(4,5)],6)
 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,3,5,7],[2,4,6,8]]
 => 1010101 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,6,8],[2,4,7],[5]]
 => 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,8],[2,4,6],[7]]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,6,7],[2,4,8],[5]]
 => 1011001 => [1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,7],[2,6,8],[4]]
 => 1010101 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,5,7],[3,6,8],[4]]
 => 0110101 => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,7],[2,4,8],[6]]
 => 1010101 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,7],[2,5,8],[6]]
 => 1001101 => [1,3,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,4,7],[3,5,8],[6]]
 => 0101101 => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,7],[2,4,6],[8]]
 => 1010101 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,6],[2,4,7],[8]]
 => 1010011 => [1,2,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,6],[2,5,7],[8]]
 => 1001011 => [1,3,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,4,6],[3,5,7],[8]]
 => 0101011 => [2,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,6,8],[2,4],[5,7]]
 => 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,8],[2,6],[4,7]]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,5,8],[3,6],[4,7]]
 => 0110110 => [2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,6,7],[2,4],[5,8]]
 => 1011001 => [1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,7],[2,6],[4,8]]
 => 1010101 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,5,7],[3,6],[4,8]]
 => 0110101 => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,7],[2,4],[6,8]]
 => 1010101 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,7],[2,5],[6,8]]
 => 1001101 => [1,3,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,4,7],[3,5],[6,8]]
 => 0101101 => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,6],[2,7],[4,8]]
 => 1010011 => [1,2,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,5,6],[3,7],[4,8]]
 => 0110011 => [2,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,6],[2,7],[5,8]]
 => 1001011 => [1,3,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,4,6],[3,7],[5,8]]
 => 0101011 => [2,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,3,6],[4,7],[5,8]]
 => 0011011 => [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,7,8],[2,4],[5],[6]]
 => 1011100 => [1,2,1,1,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,6,8],[2,7],[4],[5]]
 => 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,6,8],[3,7],[4],[5]]
 => 0111010 => [2,1,1,2,2] => ([(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,6,8],[2,4],[5],[7]]
 => 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,8],[2,6],[4],[7]]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,5,8],[3,6],[4],[7]]
 => 0110110 => [2,1,2,1,2] => ([(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,8],[2,4],[6],[7]]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,8],[2,5],[6],[7]]
 => 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,4,8],[3,5],[6],[7]]
 => 0101110 => [2,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,6,7],[2,8],[4],[5]]
 => 1011001 => [1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,6,7],[3,8],[4],[5]]
 => 0111001 => [2,1,1,3,1] => ([(0,7),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,7],[2,8],[4],[6]]
 => 1010101 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,5,7],[3,8],[4],[6]]
 => 0110101 => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,7],[2,8],[5],[6]]
 => 1001101 => [1,3,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,4,7],[3,8],[5],[6]]
 => 0101101 => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,3,7],[4,8],[5],[6]]
 => 0011101 => [3,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,6,7],[2,4],[5],[8]]
 => 1011001 => [1,2,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,7],[2,6],[4],[8]]
 => 1010101 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,5,7],[3,6],[4],[8]]
 => 0110101 => [2,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,7],[2,4],[6],[8]]
 => 1010101 => [1,2,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,4,7],[2,5],[6],[8]]
 => 1001101 => [1,3,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,2,4,7],[3,5],[6],[8]]
 => 0101101 => [2,2,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5
[[1,3,5,6],[2,7],[4],[8]]
 => 1010011 => [1,2,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5

Description

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.

Matching statistic: St000098

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 83%

Values

[[1]]
 =>  => [1] => ([],1)
 => 1
[[1,2]]
 => 0 => [2] => ([],2)
 => 1
[[1],[2]]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[[1,2,3]]
 => 00 => [3] => ([],3)
 => 1
[[1,3],[2]]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[[1,2],[3]]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[[1,2,3,4]]
 => 000 => [4] => ([],4)
 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => ([(2,3)],4)
 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => ([],5)
 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => ([(3,4)],5)
 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => ([(2,4),(3,4)],5)
 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => ([],6)
 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => ([(4,5)],6)
 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => ([(3,5),(4,5)],6)
 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[1,2,3,4,5,6,7,8]]
 => 0000000 => [8] => ([],8)
 => ? = 1
[[1,3,4,5,6,7,8],[2]]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? = 2
[[1,2,4,5,6,7,8],[3]]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2
[[1,2,3,5,6,7,8],[4]]
 => 0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,2,3,4,6,7,8],[5]]
 => 0001000 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,2,3,4,5,7,8],[6]]
 => 0000100 => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,2,3,4,5,6,8],[7]]
 => 0000010 => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,3,5,6,7,8],[2,4]]
 => 1010000 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,5,6,7,8],[3,4]]
 => 0100000 => [2,6] => ([(5,7),(6,7)],8)
 => ? = 2
[[1,3,4,6,7,8],[2,5]]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,4,6,7,8],[3,5]]
 => 0101000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,6,7,8],[4,5]]
 => 0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,3,4,5,7,8],[2,6]]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,4,5,7,8],[3,6]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,5,7,8],[4,6]]
 => 0010100 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,4,7,8],[5,6]]
 => 0001000 => [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,3,4,5,6,8],[2,7]]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,4,5,6,8],[3,7]]
 => 0100010 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,5,6,8],[4,7]]
 => 0010010 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,4,6,8],[5,7]]
 => 0001010 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,4,5,8],[6,7]]
 => 0000100 => [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,2,3,4,5,6],[7,8]]
 => 0000010 => [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,4,5,6,7,8],[2],[3]]
 => 1100000 => [1,1,6] => ([(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,5,6,7,8],[2],[4]]
 => 1010000 => [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,5,6,7,8],[3],[4]]
 => 0110000 => [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,4,6,7,8],[2],[5]]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,4,6,7,8],[3],[5]]
 => 0101000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,6,7,8],[4],[5]]
 => 0011000 => [3,1,4] => ([(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,4,5,7,8],[2],[6]]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,4,5,7,8],[3],[6]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,5,7,8],[4],[6]]
 => 0010100 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,4,7,8],[5],[6]]
 => 0001100 => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,4,5,6,8],[2],[7]]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,4,5,6,8],[3],[7]]
 => 0100010 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,5,6,8],[4],[7]]
 => 0010010 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,4,6,8],[5],[7]]
 => 0001010 => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,4,5,8],[6],[7]]
 => 0000110 => [5,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,5,7,8],[2,4,6]]
 => 1010100 => [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,2,5,7,8],[3,4,6]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,4,7,8],[2,5,6]]
 => 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,4,7,8],[3,5,6]]
 => 0101000 => [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,7,8],[4,5,6]]
 => 0010000 => [3,5] => ([(4,7),(5,7),(6,7)],8)
 => ? = 2
[[1,3,5,6,8],[2,4,7]]
 => 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,2,5,6,8],[3,4,7]]
 => 0100010 => [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,4,6,8],[2,5,7]]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,2,4,6,8],[3,5,7]]
 => 0101010 => [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[[1,2,3,6,8],[4,5,7]]
 => 0010010 => [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,3,4,5,8],[2,6,7]]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,4,5,8],[3,6,7]]
 => 0100100 => [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[[1,2,3,5,8],[4,6,7]]
 => 0010100 => [3,2,3] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000011

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 92%

Values

[[1]]
 =>  => [1] => [1,0]
 => 1
[[1,2]]
 => 0 => [2] => [1,1,0,0]
 => 1
[[1],[2]]
 => 1 => [1,1] => [1,0,1,0]
 => 2
[[1,2,3]]
 => 00 => [3] => [1,1,1,0,0,0]
 => 1
[[1,3],[2]]
 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2
[[1,2],[3]]
 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[[1,2,3,4]]
 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,3,4,5,7,8],[2,6]]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,4,5,7,8],[3,6]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,5,7,8],[4,6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,5,6,8],[4,7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
[[1,2,3,4,6,7],[5,8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[[1,2,3,4,5,7],[6,8]]
 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[[1,2,3,6,7,8],[4],[5]]
 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[[1,3,4,5,7,8],[2],[6]]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,4,5,7,8],[3],[6]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,5,7,8],[4],[6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,4,7,8],[5],[6]]
 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,5,6,8],[4],[7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
[[1,2,3,4,5,8],[6],[7]]
 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 3
[[1,2,3,4,6,7],[5],[8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[[1,2,3,4,5,7],[6],[8]]
 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[[1,3,5,7,8],[2,4,6]]
 => 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4
[[1,2,5,7,8],[3,4,6]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,3,5,6,8],[2,4,7]]
 => 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 4
[[1,2,3,6,8],[4,5,7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
[[1,3,4,5,8],[2,6,7]]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,4,5,8],[3,6,7]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,5,8],[4,6,7]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,3,5,6,7],[2,4,8]]
 => 1010001 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 4
[[1,3,4,6,7],[2,5,8]]
 => 1001001 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
[[1,2,4,6,7],[3,5,8]]
 => 0101001 => [2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
[[1,3,4,5,7],[2,6,8]]
 => 1000101 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,4,5,7],[3,6,8]]
 => 0100101 => [2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3,5,7],[4,6,8]]
 => 0010101 => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4
[[1,2,3,4,7],[5,6,8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[[1,2,3,5,6],[4,7,8]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
[[1,4,6,7,8],[2,5],[3]]
 => 1101000 => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[[1,3,6,7,8],[2,4],[5]]
 => 1011000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4
[[1,4,5,7,8],[2,6],[3]]
 => 1100100 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 4
[[1,3,5,7,8],[2,6],[4]]
 => 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4
[[1,2,5,7,8],[3,6],[4]]
 => 0110100 => [2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4
[[1,2,3,7,8],[4,6],[5]]
 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[[1,3,5,7,8],[2,4],[6]]
 => 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4
[[1,2,5,7,8],[3,4],[6]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,3,4,7,8],[2,5],[6]]
 => 1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[[1,2,4,7,8],[3,5],[6]]
 => 0101100 => [2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4
[[1,2,3,7,8],[4,5],[6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,3,5,6,8],[2,7],[4]]
 => 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 4
[[1,2,5,6,8],[3,7],[4]]
 => 0110010 => [2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 4
[[1,2,3,6,8],[4,7],[5]]
 => 0011010 => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 4
[[1,3,4,5,8],[2,7],[6]]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,4,5,8],[3,7],[6]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,5,8],[4,7],[6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,4,8],[5,7],[6]]
 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[[1,3,5,6,8],[2,4],[7]]
 => 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 4
[[1,2,3,6,8],[4,5],[7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000306

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 75%

Values

[[1]]
 =>  => [1] => [1,0]
 => 0 = 1 - 1
[[1,2]]
 => 0 => [2] => [1,1,0,0]
 => 0 = 1 - 1
[[1],[2]]
 => 1 => [1,1] => [1,0,1,0]
 => 1 = 2 - 1
[[1,2,3]]
 => 00 => [3] => [1,1,1,0,0,0]
 => 0 = 1 - 1
[[1,3],[2]]
 => 10 => [1,2] => [1,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2],[3]]
 => 01 => [2,1] => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[1],[2],[3]]
 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
[[1,2,3,4]]
 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[[1,3,4],[2]]
 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[1,2,4],[3]]
 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2,3],[4]]
 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[[1,3],[2,4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[[1,2],[3,4]]
 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3],[2],[4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[[1,2],[3],[4]]
 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[1,2,3,4,5]]
 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[1,2,4,5],[3]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[1,2,3,5],[4]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2,3,4],[5]]
 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,5],[3,4]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[[1,2,3],[4,5]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 3 - 1
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 3 - 1
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 3 - 1
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3 = 4 - 1
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 4 - 1
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 4 - 1
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
[[1,2,3,4,5,6]]
 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[[1,2,4,6,7],[3,5]]
 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
[[1,2,4,5,7],[3,6]]
 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,2,3,5,7],[4,6]]
 => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,2,4,6,7],[3],[5]]
 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
[[1,2,3,6,7],[4],[5]]
 => 001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3 - 1
[[1,2,4,5,7],[3],[6]]
 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,2,3,5,7],[4],[6]]
 => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,2,3,4,5],[6],[7]]
 => 000011 => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 3 - 1
[[1,3,5,7],[2,4,6]]
 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[[1,2,5,7],[3,4,6]]
 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,2,4,7],[3,5,6]]
 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
[[1,3,4,6],[2,5,7]]
 => 100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,4,6],[3,5,7]]
 => 010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,4,5],[3,6,7]]
 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,2,3,5],[4,6,7]]
 => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,4,6,7],[2,5],[3]]
 => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4 - 1
[[1,3,6,7],[2,4],[5]]
 => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[[1,2,6,7],[3,4],[5]]
 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
[[1,3,5,7],[2,6],[4]]
 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[[1,2,4,7],[3,6],[5]]
 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
[[1,2,3,7],[4,6],[5]]
 => 001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3 - 1
[[1,3,5,7],[2,4],[6]]
 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[[1,2,5,7],[3,4],[6]]
 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,3,4,7],[2,5],[6]]
 => 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 4 - 1
[[1,2,3,7],[4,5],[6]]
 => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,3,4,6],[2,7],[5]]
 => 100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,4,6],[3,7],[5]]
 => 010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,3,6],[4,7],[5]]
 => 001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,4,5],[3,7],[6]]
 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,2,3,5],[4,7],[6]]
 => 001010 => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3 - 1
[[1,3,4,6],[2,5],[7]]
 => 100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,4,6],[3,5],[7]]
 => 010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,3,5],[4,6],[7]]
 => 001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4 - 1
[[1,5,6,7],[2],[3],[4]]
 => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 4 - 1
[[1,4,6,7],[2],[3],[5]]
 => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4 - 1
[[1,3,6,7],[2],[4],[5]]
 => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 4 - 1
[[1,3,5,7],[2],[4],[6]]
 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[[1,3,4,7],[2],[5],[6]]
 => 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? = 4 - 1
[[1,3,4,6],[2],[5],[7]]
 => 100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,4,6],[3],[5],[7]]
 => 010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,3,6],[4],[5],[7]]
 => 001101 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,2,3,5],[4],[6],[7]]
 => 001011 => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 4 - 1
[[1,2,3,4],[5],[6],[7]]
 => 000111 => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 4 - 1
[[1,3,6],[2,4,7],[5]]
 => 101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 5 - 1
[[1,2,6],[3,4,7],[5]]
 => 010101 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 4 - 1
[[1,3,5],[2,6,7],[4]]
 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[[1,2,4],[3,6,7],[5]]
 => 010100 => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 - 1
[[1,2,3],[4,6,7],[5]]
 => 001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3 - 1
[[1,3,5],[2,4,7],[6]]
 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4 - 1
[[1,2,5],[3,4,7],[6]]
 => 010010 => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3 - 1

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

Matching statistic: St000662

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 83%

Values

[[1]]
 => [1] => [1] => [1] => 0 = 1 - 1
[[1,2]]
 => [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[[1],[2]]
 => [2,1] => [2,1] => [2,1] => 1 = 2 - 1
[[1,2,3]]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3],[2]]
 => [2,1,3] => [2,1,3] => [2,1,3] => 1 = 2 - 1
[[1,2],[3]]
 => [3,1,2] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[[1,2,4],[3]]
 => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1 = 2 - 1
[[1,2,3],[4]]
 => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 1 = 2 - 1
[[1,3],[2,4]]
 => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2 = 3 - 1
[[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => [1,4,2,3] => 1 = 2 - 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,2,4,1] => [4,2,1,3] => 2 = 3 - 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,4,2,1] => [4,3,1,2] => 2 = 3 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1 = 2 - 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 1 = 2 - 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [2,3,4,1,5] => [4,1,2,3,5] => 1 = 2 - 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [2,3,4,5,1] => [5,1,2,3,4] => 1 = 2 - 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [3,1,4,2,5] => [3,4,1,2,5] => 2 = 3 - 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,1,2,5] => [1,4,2,3,5] => 1 = 2 - 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [3,1,4,5,2] => [3,5,1,2,4] => 2 = 3 - 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [3,4,1,5,2] => [4,5,1,2,3] => 2 = 3 - 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [3,4,5,1,2] => [1,5,2,3,4] => 1 = 2 - 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2 = 3 - 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [3,2,4,1,5] => [4,2,1,3,5] => 2 = 3 - 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [3,4,2,1,5] => [4,3,1,2,5] => 2 = 3 - 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [3,2,4,5,1] => [5,2,1,3,4] => 2 = 3 - 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [3,4,2,5,1] => [5,3,1,2,4] => 2 = 3 - 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [3,4,5,2,1] => [5,4,1,2,3] => 2 = 3 - 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [4,2,1,5,3] => [4,3,5,1,2] => 3 = 4 - 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,2,5,1,3] => [1,4,5,2,3] => 2 = 3 - 1
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,2,3] => 2 = 3 - 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [4,2,5,3,1] => [5,3,4,1,2] => 3 = 4 - 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [4,5,2,3,1] => [5,1,4,2,3] => 2 = 3 - 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 3 = 4 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [4,3,2,5,1] => [5,3,2,1,4] => 3 = 4 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [4,3,5,2,1] => [5,4,2,1,3] => 3 = 4 - 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [4,5,3,2,1] => [5,4,3,1,2] => 3 = 4 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 4 = 5 - 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1 = 2 - 1
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [2,3,1,4,5,6] => [3,1,2,4,5,6] => 1 = 2 - 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [2,3,4,1,5,6] => [4,1,2,3,5,6] => 1 = 2 - 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [2,3,4,5,1,6] => [5,1,2,3,4,6] => 1 = 2 - 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 1 = 2 - 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [3,1,4,2,5,6] => [3,4,1,2,5,6] => 2 = 3 - 1
[[1,3,4,6,7],[2,5]]
 => [2,5,1,3,4,6,7] => [3,1,4,5,2,6,7] => [3,5,1,2,4,6,7] => ? = 3 - 1
[[1,2,4,6,7],[3,5]]
 => [3,5,1,2,4,6,7] => [3,4,1,5,2,6,7] => [4,5,1,2,3,6,7] => ? = 3 - 1
[[1,2,3,6,7],[4,5]]
 => [4,5,1,2,3,6,7] => [3,4,5,1,2,6,7] => [1,5,2,3,4,6,7] => ? = 2 - 1
[[1,2,3,4,7],[5,6]]
 => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => [1,6,2,3,4,5,7] => ? = 2 - 1
[[1,3,4,5,6],[2,7]]
 => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => [3,7,1,2,4,5,6] => ? = 3 - 1
[[1,2,4,5,6],[3,7]]
 => [3,7,1,2,4,5,6] => [3,4,1,5,6,7,2] => [4,7,1,2,3,5,6] => ? = 3 - 1
[[1,2,3,5,6],[4,7]]
 => [4,7,1,2,3,5,6] => [3,4,5,1,6,7,2] => [5,7,1,2,3,4,6] => ? = 3 - 1
[[1,3,5,6,7],[2],[4]]
 => [4,2,1,3,5,6,7] => [3,2,4,1,5,6,7] => [4,2,1,3,5,6,7] => ? = 3 - 1
[[1,2,5,6,7],[3],[4]]
 => [4,3,1,2,5,6,7] => [3,4,2,1,5,6,7] => [4,3,1,2,5,6,7] => ? = 3 - 1
[[1,3,4,6,7],[2],[5]]
 => [5,2,1,3,4,6,7] => [3,2,4,5,1,6,7] => [5,2,1,3,4,6,7] => ? = 3 - 1
[[1,2,4,6,7],[3],[5]]
 => [5,3,1,2,4,6,7] => [3,4,2,5,1,6,7] => [5,3,1,2,4,6,7] => ? = 3 - 1
[[1,2,3,6,7],[4],[5]]
 => [5,4,1,2,3,6,7] => [3,4,5,2,1,6,7] => [5,4,1,2,3,6,7] => ? = 3 - 1
[[1,3,4,5,7],[2],[6]]
 => [6,2,1,3,4,5,7] => [3,2,4,5,6,1,7] => [6,2,1,3,4,5,7] => ? = 3 - 1
[[1,2,4,5,7],[3],[6]]
 => [6,3,1,2,4,5,7] => [3,4,2,5,6,1,7] => [6,3,1,2,4,5,7] => ? = 3 - 1
[[1,2,3,5,7],[4],[6]]
 => [6,4,1,2,3,5,7] => [3,4,5,2,6,1,7] => [6,4,1,2,3,5,7] => ? = 3 - 1
[[1,2,3,4,7],[5],[6]]
 => [6,5,1,2,3,4,7] => [3,4,5,6,2,1,7] => [6,5,1,2,3,4,7] => ? = 3 - 1
[[1,2,5,7],[3,4,6]]
 => [3,4,6,1,2,5,7] => [4,5,1,2,6,3,7] => [2,5,6,1,3,4,7] => ? = 3 - 1
[[1,3,4,7],[2,5,6]]
 => [2,5,6,1,3,4,7] => [4,1,5,6,2,3,7] => [3,1,6,2,4,5,7] => ? = 3 - 1
[[1,3,5,6],[2,4,7]]
 => [2,4,7,1,3,5,6] => [4,1,5,2,6,7,3] => [4,5,7,1,2,3,6] => ? = 4 - 1
[[1,3,4,6],[2,5,7]]
 => [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => [4,6,7,1,2,3,5] => ? = 4 - 1
[[1,3,4,5],[2,6,7]]
 => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => [3,1,7,2,4,5,6] => ? = 3 - 1
[[1,2,4,5],[3,6,7]]
 => [3,6,7,1,2,4,5] => [4,5,1,6,7,2,3] => [4,1,7,2,3,5,6] => ? = 3 - 1
[[1,2,3,5],[4,6,7]]
 => [4,6,7,1,2,3,5] => [4,5,6,1,7,2,3] => [5,1,7,2,3,4,6] => ? = 3 - 1
[[1,4,6,7],[2,5],[3]]
 => [3,2,5,1,4,6,7] => [4,2,1,5,3,6,7] => [4,3,5,1,2,6,7] => ? = 4 - 1
[[1,2,6,7],[3,5],[4]]
 => [4,3,5,1,2,6,7] => [4,5,2,1,3,6,7] => [4,1,5,2,3,6,7] => ? = 3 - 1
[[1,3,6,7],[2,4],[5]]
 => [5,2,4,1,3,6,7] => [4,2,5,3,1,6,7] => [5,3,4,1,2,6,7] => ? = 4 - 1
[[1,2,6,7],[3,4],[5]]
 => [5,3,4,1,2,6,7] => [4,5,2,3,1,6,7] => [5,1,4,2,3,6,7] => ? = 3 - 1
[[1,4,5,7],[2,6],[3]]
 => [3,2,6,1,4,5,7] => [4,2,1,5,6,3,7] => [4,3,6,1,2,5,7] => ? = 4 - 1
[[1,3,5,7],[2,6],[4]]
 => [4,2,6,1,3,5,7] => [4,2,5,1,6,3,7] => [5,3,6,1,2,4,7] => ? = 4 - 1
[[1,2,5,7],[3,6],[4]]
 => [4,3,6,1,2,5,7] => [4,5,2,1,6,3,7] => [5,4,6,1,2,3,7] => ? = 4 - 1
[[1,2,4,7],[3,6],[5]]
 => [5,3,6,1,2,4,7] => [4,5,2,6,1,3,7] => [1,5,6,2,3,4,7] => ? = 3 - 1
[[1,3,5,7],[2,4],[6]]
 => [6,2,4,1,3,5,7] => [4,2,5,3,6,1,7] => [6,3,4,1,2,5,7] => ? = 4 - 1
[[1,2,5,7],[3,4],[6]]
 => [6,3,4,1,2,5,7] => [4,5,2,3,6,1,7] => [6,1,4,2,3,5,7] => ? = 3 - 1
[[1,3,4,7],[2,5],[6]]
 => [6,2,5,1,3,4,7] => [4,2,5,6,3,1,7] => [6,3,5,1,2,4,7] => ? = 4 - 1
[[1,2,4,7],[3,5],[6]]
 => [6,3,5,1,2,4,7] => [4,5,2,6,3,1,7] => [6,4,5,1,2,3,7] => ? = 4 - 1
[[1,2,3,7],[4,5],[6]]
 => [6,4,5,1,2,3,7] => [4,5,6,2,3,1,7] => [6,1,5,2,3,4,7] => ? = 3 - 1
[[1,4,5,6],[2,7],[3]]
 => [3,2,7,1,4,5,6] => [4,2,1,5,6,7,3] => [4,3,7,1,2,5,6] => ? = 4 - 1
[[1,3,5,6],[2,7],[4]]
 => [4,2,7,1,3,5,6] => [4,2,5,1,6,7,3] => [5,3,7,1,2,4,6] => ? = 4 - 1
[[1,2,5,6],[3,7],[4]]
 => [4,3,7,1,2,5,6] => [4,5,2,1,6,7,3] => [5,4,7,1,2,3,6] => ? = 4 - 1
[[1,3,4,6],[2,7],[5]]
 => [5,2,7,1,3,4,6] => [4,2,5,6,1,7,3] => [6,3,7,1,2,4,5] => ? = 4 - 1
[[1,2,4,6],[3,7],[5]]
 => [5,3,7,1,2,4,6] => [4,5,2,6,1,7,3] => [6,4,7,1,2,3,5] => ? = 4 - 1
[[1,2,3,6],[4,7],[5]]
 => [5,4,7,1,2,3,6] => [4,5,6,2,1,7,3] => [6,5,7,1,2,3,4] => ? = 4 - 1
[[1,2,3,4],[5,7],[6]]
 => [6,5,7,1,2,3,4] => [4,5,6,7,2,1,3] => [6,1,7,2,3,4,5] => ? = 3 - 1
[[1,2,5,6],[3,4],[7]]
 => [7,3,4,1,2,5,6] => [4,5,2,3,6,7,1] => [7,1,4,2,3,5,6] => ? = 3 - 1
[[1,3,4,6],[2,5],[7]]
 => [7,2,5,1,3,4,6] => [4,2,5,6,3,7,1] => [7,3,5,1,2,4,6] => ? = 4 - 1
[[1,2,4,6],[3,5],[7]]
 => [7,3,5,1,2,4,6] => [4,5,2,6,3,7,1] => [7,4,5,1,2,3,6] => ? = 4 - 1
[[1,2,3,6],[4,5],[7]]
 => [7,4,5,1,2,3,6] => [4,5,6,2,3,7,1] => [7,1,5,2,3,4,6] => ? = 3 - 1
[[1,3,4,5],[2,6],[7]]
 => [7,2,6,1,3,4,5] => [4,2,5,6,7,3,1] => [7,3,6,1,2,4,5] => ? = 4 - 1
[[1,2,4,5],[3,6],[7]]
 => [7,3,6,1,2,4,5] => [4,5,2,6,7,3,1] => [7,4,6,1,2,3,5] => ? = 4 - 1
[[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => [4,5,6,7,2,3,1] => [7,1,6,2,3,4,5] => ? = 3 - 1

Description

The staircase size of the code of a permutation. The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ . The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ . This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.

Matching statistic: St001068
(load all 2 compositions to match this statistic)

Mapping

Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 58%

Values

[[1]]
 =>  => [1] => [1,0]
 => 1
[[1,2]]
 => 0 => [2] => [1,1,0,0]
 => 1
[[1],[2]]
 => 1 => [1,1] => [1,0,1,0]
 => 2
[[1,2,3]]
 => 00 => [3] => [1,1,1,0,0,0]
 => 1
[[1,3],[2]]
 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2
[[1,2],[3]]
 => 01 => [2,1] => [1,1,0,0,1,0]
 => 2
[[1],[2],[3]]
 => 11 => [1,1,1] => [1,0,1,0,1,0]
 => 3
[[1,2,3,4]]
 => 000 => [4] => [1,1,1,1,0,0,0,0]
 => 1
[[1,3,4],[2]]
 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[[1,2,4],[3]]
 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,2,3],[4]]
 => 001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2
[[1,3],[2,4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[[1,2],[3,4]]
 => 010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[[1,4],[2],[3]]
 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[[1,3],[2],[4]]
 => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3
[[1,2],[3],[4]]
 => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3
[[1],[2],[3],[4]]
 => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[[1,2,3,4,5]]
 => 0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[[1,3,4,5],[2]]
 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[[1,2,4,5],[3]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,2,3,5],[4]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4],[5]]
 => 0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2
[[1,3,5],[2,4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3,4]]
 => 0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[[1,3,4],[2,5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[1,2,4],[3,5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,3],[4,5]]
 => 0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2
[[1,4,5],[2],[3]]
 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[[1,3,5],[2],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2,5],[3],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[1,3,4],[2],[5]]
 => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3
[[1,2,4],[3],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,2,3],[4],[5]]
 => 0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[1,4],[2,5],[3]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[[1,3],[2,5],[4]]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3
[[1,2],[3,5],[4]]
 => 0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[1,3],[2,4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[[1,2],[3,4],[5]]
 => 0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 3
[[1,5],[2],[3],[4]]
 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[1,4],[2],[3],[5]]
 => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4
[[1,3],[2],[4],[5]]
 => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 4
[[1,2],[3],[4],[5]]
 => 0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4
[[1],[2],[3],[4],[5]]
 => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,2,3,4,5,6]]
 => 00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[[1,3,4,5,6],[2]]
 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
[[1,2,4,5,6],[3]]
 => 01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[[1,2,3,5,6],[4]]
 => 00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2
[[1,2,3,4,6],[5]]
 => 00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[[1,2,3,4,5],[6]]
 => 00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
[[1,3,5,6],[2,4]]
 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 3
[[1,2,3,4,5,6,7,8]]
 => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
[[1,3,4,5,6,7,8],[2]]
 => 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 2
[[1,2,4,5,6,7,8],[3]]
 => 0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[[1,2,3,5,6,7,8],[4]]
 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[[1,2,3,4,6,7,8],[5]]
 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[[1,2,3,4,5,7,8],[6]]
 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[[1,2,3,4,5,6,8],[7]]
 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[[1,2,3,4,5,6,7],[8]]
 => 0000001 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 2
[[1,3,5,6,7,8],[2,4]]
 => 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[[1,2,5,6,7,8],[3,4]]
 => 0100000 => [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[[1,3,4,6,7,8],[2,5]]
 => 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[[1,2,4,6,7,8],[3,5]]
 => 0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[[1,2,3,6,7,8],[4,5]]
 => 0010000 => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[[1,3,4,5,7,8],[2,6]]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,4,5,7,8],[3,6]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,5,7,8],[4,6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,4,7,8],[5,6]]
 => 0001000 => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[[1,3,4,5,6,8],[2,7]]
 => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 3
[[1,2,4,5,6,8],[3,7]]
 => 0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3
[[1,2,3,5,6,8],[4,7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
[[1,2,3,4,6,8],[5,7]]
 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[[1,2,3,4,5,8],[6,7]]
 => 0000100 => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[[1,3,4,5,6,7],[2,8]]
 => 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 3
[[1,2,4,5,6,7],[3,8]]
 => 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 3
[[1,2,3,5,6,7],[4,8]]
 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[[1,2,3,4,6,7],[5,8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[[1,2,3,4,5,7],[6,8]]
 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[[1,2,3,4,5,6],[7,8]]
 => 0000010 => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 2
[[1,4,5,6,7,8],[2],[3]]
 => 1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 3
[[1,3,5,6,7,8],[2],[4]]
 => 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[[1,2,5,6,7,8],[3],[4]]
 => 0110000 => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 3
[[1,3,4,6,7,8],[2],[5]]
 => 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[[1,2,4,6,7,8],[3],[5]]
 => 0101000 => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[[1,2,3,6,7,8],[4],[5]]
 => 0011000 => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[[1,3,4,5,7,8],[2],[6]]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,4,5,7,8],[3],[6]]
 => 0100100 => [2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,5,7,8],[4],[6]]
 => 0010100 => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
[[1,2,3,4,7,8],[5],[6]]
 => 0001100 => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[[1,3,4,5,6,8],[2],[7]]
 => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 3
[[1,2,4,5,6,8],[3],[7]]
 => 0100010 => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3
[[1,2,3,5,6,8],[4],[7]]
 => 0010010 => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 3
[[1,2,3,4,6,8],[5],[7]]
 => 0001010 => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[[1,2,3,4,5,8],[6],[7]]
 => 0000110 => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 3
[[1,3,4,5,6,7],[2],[8]]
 => 1000001 => [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 3
[[1,2,4,5,6,7],[3],[8]]
 => 0100001 => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 3
[[1,2,3,5,6,7],[4],[8]]
 => 0010001 => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[[1,2,3,4,6,7],[5],[8]]
 => 0001001 => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 3
[[1,2,3,4,5,7],[6],[8]]
 => 0000101 => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[[1,2,3,4,5,6],[7],[8]]
 => 0000011 => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 3
[[1,3,5,7,8],[2,4,6]]
 => 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 4

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001494The Alon-Tarsi number of a graph. St000053The number of valleys of the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000245The number of ascents of a permutation. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000672The number of minimal elements in Bruhat order not less than the permutation. St001812The biclique partition number of a graph. St000167The number of leaves of an ordered tree. St000354The number of recoils of a permutation. St000703The number of deficiencies of a permutation. St000470The number of runs in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000021The number of descents of a permutation. St000015The number of peaks of a Dyck path. St000822The Hadwiger number of the graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St000325The width of the tree associated to a permutation. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000083The number of left oriented leafs of a binary tree except the first one. St000159The number of distinct parts of the integer partition. St001427The number of descents of a signed permutation. St001330The hat guessing number of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000732The number of double deficiencies of a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001727The number of invisible inversions of a permutation. St001896The number of right descents of a signed permutations.