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Your data matches 103 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1
[1,1] => 11 => 2
[2] => 10 => 1
[1,1,1] => 111 => 3
[1,2] => 110 => 2
[2,1] => 101 => 1
[3] => 100 => 1
[1,1,1,1] => 1111 => 4
[1,1,2] => 1110 => 3
[1,2,1] => 1101 => 2
[1,3] => 1100 => 2
[2,1,1] => 1011 => 1
[2,2] => 1010 => 1
[3,1] => 1001 => 1
[4] => 1000 => 1
[1,1,1,1,1] => 11111 => 5
[1,1,1,2] => 11110 => 4
[1,1,2,1] => 11101 => 3
[1,1,3] => 11100 => 3
[1,2,1,1] => 11011 => 2
[1,2,2] => 11010 => 2
[1,3,1] => 11001 => 2
[1,4] => 11000 => 2
[2,1,1,1] => 10111 => 1
[2,1,2] => 10110 => 1
[2,2,1] => 10101 => 1
[2,3] => 10100 => 1
[3,1,1] => 10011 => 1
[3,2] => 10010 => 1
[4,1] => 10001 => 1
[5] => 10000 => 1
[1,1,1,1,1,1] => 111111 => 6
[1,1,1,1,2] => 111110 => 5
[1,1,1,2,1] => 111101 => 4
[1,1,1,3] => 111100 => 4
[1,1,2,1,1] => 111011 => 3
[1,1,2,2] => 111010 => 3
[1,1,3,1] => 111001 => 3
[1,1,4] => 111000 => 3
[1,2,1,1,1] => 110111 => 2
[1,2,1,2] => 110110 => 2
[1,2,2,1] => 110101 => 2
[1,2,3] => 110100 => 2
[1,3,1,1] => 110011 => 2
[1,3,2] => 110010 => 2
[1,4,1] => 110001 => 2
[1,5] => 110000 => 2
[2,1,1,1,1] => 101111 => 1
[2,1,1,2] => 101110 => 1
[2,1,2,1] => 101101 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 => 2 = 1 + 1
[1,1] => 11 => 00 => 3 = 2 + 1
[2] => 10 => 01 => 2 = 1 + 1
[1,1,1] => 111 => 000 => 4 = 3 + 1
[1,2] => 110 => 001 => 3 = 2 + 1
[2,1] => 101 => 010 => 2 = 1 + 1
[3] => 100 => 011 => 2 = 1 + 1
[1,1,1,1] => 1111 => 0000 => 5 = 4 + 1
[1,1,2] => 1110 => 0001 => 4 = 3 + 1
[1,2,1] => 1101 => 0010 => 3 = 2 + 1
[1,3] => 1100 => 0011 => 3 = 2 + 1
[2,1,1] => 1011 => 0100 => 2 = 1 + 1
[2,2] => 1010 => 0101 => 2 = 1 + 1
[3,1] => 1001 => 0110 => 2 = 1 + 1
[4] => 1000 => 0111 => 2 = 1 + 1
[1,1,1,1,1] => 11111 => 00000 => 6 = 5 + 1
[1,1,1,2] => 11110 => 00001 => 5 = 4 + 1
[1,1,2,1] => 11101 => 00010 => 4 = 3 + 1
[1,1,3] => 11100 => 00011 => 4 = 3 + 1
[1,2,1,1] => 11011 => 00100 => 3 = 2 + 1
[1,2,2] => 11010 => 00101 => 3 = 2 + 1
[1,3,1] => 11001 => 00110 => 3 = 2 + 1
[1,4] => 11000 => 00111 => 3 = 2 + 1
[2,1,1,1] => 10111 => 01000 => 2 = 1 + 1
[2,1,2] => 10110 => 01001 => 2 = 1 + 1
[2,2,1] => 10101 => 01010 => 2 = 1 + 1
[2,3] => 10100 => 01011 => 2 = 1 + 1
[3,1,1] => 10011 => 01100 => 2 = 1 + 1
[3,2] => 10010 => 01101 => 2 = 1 + 1
[4,1] => 10001 => 01110 => 2 = 1 + 1
[5] => 10000 => 01111 => 2 = 1 + 1
[1,1,1,1,1,1] => 111111 => 000000 => 7 = 6 + 1
[1,1,1,1,2] => 111110 => 000001 => 6 = 5 + 1
[1,1,1,2,1] => 111101 => 000010 => 5 = 4 + 1
[1,1,1,3] => 111100 => 000011 => 5 = 4 + 1
[1,1,2,1,1] => 111011 => 000100 => 4 = 3 + 1
[1,1,2,2] => 111010 => 000101 => 4 = 3 + 1
[1,1,3,1] => 111001 => 000110 => 4 = 3 + 1
[1,1,4] => 111000 => 000111 => 4 = 3 + 1
[1,2,1,1,1] => 110111 => 001000 => 3 = 2 + 1
[1,2,1,2] => 110110 => 001001 => 3 = 2 + 1
[1,2,2,1] => 110101 => 001010 => 3 = 2 + 1
[1,2,3] => 110100 => 001011 => 3 = 2 + 1
[1,3,1,1] => 110011 => 001100 => 3 = 2 + 1
[1,3,2] => 110010 => 001101 => 3 = 2 + 1
[1,4,1] => 110001 => 001110 => 3 = 2 + 1
[1,5] => 110000 => 001111 => 3 = 2 + 1
[2,1,1,1,1] => 101111 => 010000 => 2 = 1 + 1
[2,1,1,2] => 101110 => 010001 => 2 = 1 + 1
[2,1,2,1] => 101101 => 010010 => 2 = 1 + 1
[1,3,1,4,1] => 1100110001 => 0011001110 => ? = 2 + 1
[1,3,2,3,1] => 1100101001 => 0011010110 => ? = 2 + 1
[1,3,3,1,2] => 1100100110 => 0011011001 => ? = 2 + 1
[1,3,3,2,1] => 1100100101 => 0011011010 => ? = 2 + 1
[1,4,3,2] => 1100010010 => 0011101101 => ? = 2 + 1
[1,5,4] => 1100001000 => 0011110111 => ? = 2 + 1
[2,1,1,5,1] => 1011100001 => 0100011110 => ? = 1 + 1
[2,1,2,4,1] => 1011010001 => 0100101110 => ? = 1 + 1
[2,1,3,3,1] => 1011001001 => 0100110110 => ? = 1 + 1
[2,2,1,4,1] => 1010110001 => 0101001110 => ? = 1 + 1
[2,3,1,3,1] => 1010011001 => ? => ? = 1 + 1
[3,1,1,1,4] => 1001111000 => 0110000111 => ? = 1 + 1
[3,1,1,2,3] => 1001110100 => 0110001011 => ? = 1 + 1
[3,1,1,4,1] => 1001110001 => 0110001110 => ? = 1 + 1
[3,1,2,1,3] => 1001101100 => 0110010011 => ? = 1 + 1
[4,1,1,1,3] => 1000111100 => 0111000011 => ? = 1 + 1
[3,2,1,1,2,3] => 100101110100 => 011010001011 => ? = 1 + 1
[3,1,1,2,2,3] => 100111010100 => 011000101011 => ? = 1 + 1
[3,1,1,1,2,4] => 100111101000 => 011000010111 => ? = 1 + 1
[4,1,1,2,1,3] => 100011101100 => 011100010011 => ? = 1 + 1
[4,1,1,1,1,4] => 100011111000 => 011100000111 => ? = 1 + 1
[1,2,2,3,3,1] => 110101001001 => 001010110110 => ? = 2 + 1
[1,3,2,3,2,1] => 110010100101 => 001101011010 => ? = 2 + 1
[1,3,2,2,3,1] => 110010101001 => 001101010110 => ? = 2 + 1
[1,2,1,1,1,1,1,1,2] => 11011111110 => 00100000001 => ? = 2 + 1
[3,3,3,2] => 10010010010 => 01101101101 => ? = 1 + 1
[3,4,4] => 10010001000 => 01101110111 => ? = 1 + 1
[2,3,3,3] => 10100100100 => ? => ? = 1 + 1
[4,4,3] => 10001000100 => 01110111011 => ? = 1 + 1
[2,1,1,1,6,1] => 101111000001 => 010000111110 => ? = 1 + 1
[2,2,1,1,1,1,1,2] => 10101111110 => 01010000001 => ? = 1 + 1
[1,2,2,4,2,1] => 110101000101 => 001010111010 => ? = 2 + 1
[1,4,2,2,2,1] => 110001010101 => 001110101010 => ? = 2 + 1
[1,2,2,2,4,1] => 110101010001 => 001010101110 => ? = 2 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => [1] => 1
[1,1] => 11 => [2] => 2
[2] => 10 => [1,1] => 1
[1,1,1] => 111 => [3] => 3
[1,2] => 110 => [2,1] => 2
[2,1] => 101 => [1,1,1] => 1
[3] => 100 => [1,2] => 1
[1,1,1,1] => 1111 => [4] => 4
[1,1,2] => 1110 => [3,1] => 3
[1,2,1] => 1101 => [2,1,1] => 2
[1,3] => 1100 => [2,2] => 2
[2,1,1] => 1011 => [1,1,2] => 1
[2,2] => 1010 => [1,1,1,1] => 1
[3,1] => 1001 => [1,2,1] => 1
[4] => 1000 => [1,3] => 1
[1,1,1,1,1] => 11111 => [5] => 5
[1,1,1,2] => 11110 => [4,1] => 4
[1,1,2,1] => 11101 => [3,1,1] => 3
[1,1,3] => 11100 => [3,2] => 3
[1,2,1,1] => 11011 => [2,1,2] => 2
[1,2,2] => 11010 => [2,1,1,1] => 2
[1,3,1] => 11001 => [2,2,1] => 2
[1,4] => 11000 => [2,3] => 2
[2,1,1,1] => 10111 => [1,1,3] => 1
[2,1,2] => 10110 => [1,1,2,1] => 1
[2,2,1] => 10101 => [1,1,1,1,1] => 1
[2,3] => 10100 => [1,1,1,2] => 1
[3,1,1] => 10011 => [1,2,2] => 1
[3,2] => 10010 => [1,2,1,1] => 1
[4,1] => 10001 => [1,3,1] => 1
[5] => 10000 => [1,4] => 1
[1,1,1,1,1,1] => 111111 => [6] => 6
[1,1,1,1,2] => 111110 => [5,1] => 5
[1,1,1,2,1] => 111101 => [4,1,1] => 4
[1,1,1,3] => 111100 => [4,2] => 4
[1,1,2,1,1] => 111011 => [3,1,2] => 3
[1,1,2,2] => 111010 => [3,1,1,1] => 3
[1,1,3,1] => 111001 => [3,2,1] => 3
[1,1,4] => 111000 => [3,3] => 3
[1,2,1,1,1] => 110111 => [2,1,3] => 2
[1,2,1,2] => 110110 => [2,1,2,1] => 2
[1,2,2,1] => 110101 => [2,1,1,1,1] => 2
[1,2,3] => 110100 => [2,1,1,2] => 2
[1,3,1,1] => 110011 => [2,2,2] => 2
[1,3,2] => 110010 => [2,2,1,1] => 2
[1,4,1] => 110001 => [2,3,1] => 2
[1,5] => 110000 => [2,4] => 2
[2,1,1,1,1] => 101111 => [1,1,4] => 1
[2,1,1,2] => 101110 => [1,1,3,1] => 1
[2,1,2,1] => 101101 => [1,1,2,1,1] => 1
[1,1,2,1,1,1,1,2] => 1110111110 => [3,1,5,1] => ? = 3
[1,1,2,5,1] => 1110100001 => [3,1,1,4,1] => ? = 3
[1,1,3,4,1] => 1110010001 => [3,2,1,3,1] => ? = 3
[1,1,4,3,1] => 1110001001 => [3,3,1,2,1] => ? = 3
[1,1,5,2,1] => 1110000101 => [3,4,1,1,1] => ? = 3
[1,2,1,5,1] => 1101100001 => [2,1,2,4,1] => ? = 2
[1,2,2,4,1] => 1101010001 => [2,1,1,1,1,3,1] => ? = 2
[1,2,3,1,1,2] => 1101001110 => [2,1,1,2,3,1] => ? = 2
[1,2,4,2,1] => 1101000101 => [2,1,1,3,1,1,1] => ? = 2
[1,3,2,3,1] => 1100101001 => [2,2,1,1,1,2,1] => ? = 2
[1,3,3,1,2] => 1100100110 => [2,2,1,2,2,1] => ? = 2
[1,4,2,2,1] => 1100010101 => [2,3,1,1,1,1,1] => ? = 2
[1,4,3,2] => 1100010010 => [2,3,1,2,1,1] => ? = 2
[2,1,1,1,1,1,2,1] => 1011111101 => [1,1,6,1,1] => ? = 1
[2,1,1,5,1] => 1011100001 => [1,1,3,4,1] => ? = 1
[2,1,3,3,1] => 1011001001 => [1,1,2,2,1,2,1] => ? = 1
[2,2,1,4,1] => 1010110001 => [1,1,1,1,2,3,1] => ? = 1
[2,2,3,1,2] => 1010100110 => [1,1,1,1,1,2,2,1] => ? = 1
[2,3,1,2,2] => 1010011010 => [1,1,1,2,2,1,1,1] => ? = 1
[2,3,1,3,1] => 1010011001 => [1,1,1,2,2,2,1] => ? = 1
[2,3,2,1,2] => 1010010110 => [1,1,1,2,1,1,2,1] => ? = 1
[2,4,1,1,2] => 1010001110 => [1,1,1,3,3,1] => ? = 1
[3,1,2,1,3] => 1001101100 => [1,2,2,1,2,2] => ? = 1
[3,1,2,2,2] => 1001101010 => [1,2,2,1,1,1,1,1] => ? = 1
[3,1,2,3,1] => 1001101001 => [1,2,2,1,1,2,1] => ? = 1
[3,1,3,2,1] => 1001100101 => [1,2,2,2,1,1,1] => ? = 1
[3,2,1,2,2] => 1001011010 => [1,2,1,1,2,1,1,1] => ? = 1
[3,2,1,3,1] => 1001011001 => [1,2,1,1,2,2,1] => ? = 1
[3,2,2,1,2] => 1001010110 => [1,2,1,1,1,1,2,1] => ? = 1
[3,3,1,1,2] => 1001001110 => [1,2,1,2,3,1] => ? = 1
[4,1,1,2,2] => 1000111010 => [1,3,3,1,1,1] => ? = 1
[4,1,2,1,2] => 1000110110 => [1,3,2,1,2,1] => ? = 1
[3,2,1,1,2,3] => 100101110100 => [1,2,1,1,3,1,1,2] => ? = 1
[3,1,1,2,2,3] => 100111010100 => [1,2,3,1,1,1,1,2] => ? = 1
[3,1,1,1,2,4] => 100111101000 => [1,2,4,1,1,3] => ? = 1
[4,1,1,2,1,3] => 100011101100 => [1,3,3,1,2,2] => ? = 1
[1,2,2,3,3,1] => 110101001001 => [2,1,1,1,1,2,1,2,1] => ? = 2
[1,2,3,3,2,1] => 110100100101 => [2,1,1,2,1,2,1,1,1] => ? = 2
[1,2,3,2,3,1] => 110100101001 => [2,1,1,2,1,1,1,2,1] => ? = 2
[1,3,3,2,2,1] => 110010010101 => [2,2,1,2,1,1,1,1,1] => ? = 2
[1,3,2,3,2,1] => 110010100101 => [2,2,1,1,1,2,1,1,1] => ? = 2
[1,3,2,2,3,1] => 110010101001 => [2,2,1,1,1,1,1,2,1] => ? = 2
[2,3,2,2,2,1] => 101001010101 => [1,1,1,2,1,1,1,1,1,1,1] => ? = 1
[2,2,3,2,2,1] => 101010010101 => [1,1,1,1,1,2,1,1,1,1,1] => ? = 1
[2,2,2,3,2,1] => 101010100101 => [1,1,1,1,1,1,1,2,1,1,1] => ? = 1
[2,2,2,2,3,1] => 101010101001 => [1,1,1,1,1,1,1,1,1,2,1] => ? = 1
[1,2,1,1,1,1,1,1,2] => 11011111110 => [2,1,7,1] => ? = 2
[3,3,3,2] => 10010010010 => [1,2,1,2,1,2,1,1] => ? = 1
[2,3,3,3] => 10100100100 => [1,1,1,2,1,2,1,2] => ? = 1
[4,4,3] => 10001000100 => [1,3,1,3,1,2] => ? = 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => [1] => [1] => 1
[1,1] => 11 => [2] => [2] => 2
[2] => 10 => [1,1] => [1,1] => 1
[1,1,1] => 111 => [3] => [3] => 3
[1,2] => 110 => [2,1] => [1,2] => 2
[2,1] => 101 => [1,1,1] => [1,1,1] => 1
[3] => 100 => [1,2] => [2,1] => 1
[1,1,1,1] => 1111 => [4] => [4] => 4
[1,1,2] => 1110 => [3,1] => [1,3] => 3
[1,2,1] => 1101 => [2,1,1] => [1,1,2] => 2
[1,3] => 1100 => [2,2] => [2,2] => 2
[2,1,1] => 1011 => [1,1,2] => [2,1,1] => 1
[2,2] => 1010 => [1,1,1,1] => [1,1,1,1] => 1
[3,1] => 1001 => [1,2,1] => [1,2,1] => 1
[4] => 1000 => [1,3] => [3,1] => 1
[1,1,1,1,1] => 11111 => [5] => [5] => 5
[1,1,1,2] => 11110 => [4,1] => [1,4] => 4
[1,1,2,1] => 11101 => [3,1,1] => [1,1,3] => 3
[1,1,3] => 11100 => [3,2] => [2,3] => 3
[1,2,1,1] => 11011 => [2,1,2] => [2,1,2] => 2
[1,2,2] => 11010 => [2,1,1,1] => [1,1,1,2] => 2
[1,3,1] => 11001 => [2,2,1] => [1,2,2] => 2
[1,4] => 11000 => [2,3] => [3,2] => 2
[2,1,1,1] => 10111 => [1,1,3] => [3,1,1] => 1
[2,1,2] => 10110 => [1,1,2,1] => [1,2,1,1] => 1
[2,2,1] => 10101 => [1,1,1,1,1] => [1,1,1,1,1] => 1
[2,3] => 10100 => [1,1,1,2] => [2,1,1,1] => 1
[3,1,1] => 10011 => [1,2,2] => [2,2,1] => 1
[3,2] => 10010 => [1,2,1,1] => [1,1,2,1] => 1
[4,1] => 10001 => [1,3,1] => [1,3,1] => 1
[5] => 10000 => [1,4] => [4,1] => 1
[1,1,1,1,1,1] => 111111 => [6] => [6] => 6
[1,1,1,1,2] => 111110 => [5,1] => [1,5] => 5
[1,1,1,2,1] => 111101 => [4,1,1] => [1,1,4] => 4
[1,1,1,3] => 111100 => [4,2] => [2,4] => 4
[1,1,2,1,1] => 111011 => [3,1,2] => [2,1,3] => 3
[1,1,2,2] => 111010 => [3,1,1,1] => [1,1,1,3] => 3
[1,1,3,1] => 111001 => [3,2,1] => [1,2,3] => 3
[1,1,4] => 111000 => [3,3] => [3,3] => 3
[1,2,1,1,1] => 110111 => [2,1,3] => [3,1,2] => 2
[1,2,1,2] => 110110 => [2,1,2,1] => [1,2,1,2] => 2
[1,2,2,1] => 110101 => [2,1,1,1,1] => [1,1,1,1,2] => 2
[1,2,3] => 110100 => [2,1,1,2] => [2,1,1,2] => 2
[1,3,1,1] => 110011 => [2,2,2] => [2,2,2] => 2
[1,3,2] => 110010 => [2,2,1,1] => [1,1,2,2] => 2
[1,4,1] => 110001 => [2,3,1] => [1,3,2] => 2
[1,5] => 110000 => [2,4] => [4,2] => 2
[2,1,1,1,1] => 101111 => [1,1,4] => [4,1,1] => 1
[2,1,1,2] => 101110 => [1,1,3,1] => [1,3,1,1] => 1
[2,1,2,1] => 101101 => [1,1,2,1,1] => [1,1,2,1,1] => 1
[1,1,1,1,1,1,1,1] => 11111111 => [8] => [8] => ? = 8
[1,1,1,1,1,2,1] => 11111101 => [6,1,1] => [1,1,6] => ? = 6
[1,1,1,1,1,3] => 11111100 => [6,2] => [2,6] => ? = 6
[1,1,1,1,2,1,1] => 11111011 => [5,1,2] => [2,1,5] => ? = 5
[1,1,1,1,2,2] => 11111010 => [5,1,1,1] => [1,1,1,5] => ? = 5
[1,1,1,1,3,1] => 11111001 => [5,2,1] => [1,2,5] => ? = 5
[1,1,1,1,4] => 11111000 => [5,3] => [3,5] => ? = 5
[1,1,1,2,1,1,1] => 11110111 => [4,1,3] => [3,1,4] => ? = 4
[1,1,1,2,2,1] => 11110101 => [4,1,1,1,1] => [1,1,1,1,4] => ? = 4
[1,1,1,2,3] => 11110100 => [4,1,1,2] => [2,1,1,4] => ? = 4
[1,1,1,3,1,1] => 11110011 => [4,2,2] => [2,2,4] => ? = 4
[1,1,1,3,2] => 11110010 => [4,2,1,1] => [1,1,2,4] => ? = 4
[1,1,1,4,1] => 11110001 => [4,3,1] => [1,3,4] => ? = 4
[1,1,2,1,1,1,1] => 11101111 => [3,1,4] => [4,1,3] => ? = 3
[1,1,2,1,2,1] => 11101101 => [3,1,2,1,1] => [1,1,2,1,3] => ? = 3
[1,1,2,2,1,1] => 11101011 => [3,1,1,1,2] => [2,1,1,1,3] => ? = 3
[1,1,2,2,2] => 11101010 => [3,1,1,1,1,1] => [1,1,1,1,1,3] => ? = 3
[1,1,2,3,1] => 11101001 => [3,1,1,2,1] => [1,2,1,1,3] => ? = 3
[1,1,3,2,1] => 11100101 => [3,2,1,1,1] => [1,1,1,2,3] => ? = 3
[1,1,6] => 11100000 => [3,5] => [5,3] => ? = 3
[1,2,1,1,1,1,1] => 11011111 => [2,1,5] => [5,1,2] => ? = 2
[1,2,2,1,1,1] => 11010111 => [2,1,1,1,3] => [3,1,1,1,2] => ? = 2
[1,2,2,2,1] => 11010101 => [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2] => ? = 2
[1,2,5] => 11010000 => [2,1,1,4] => [4,1,1,2] => ? = 2
[1,3,1,1,1,1] => 11001111 => [2,2,4] => [4,2,2] => ? = 2
[1,5,2] => 11000010 => [2,4,1,1] => [1,1,4,2] => ? = 2
[1,7] => 11000000 => [2,6] => [6,2] => ? = 2
[2,1,1,1,2,1] => 10111101 => [1,1,4,1,1] => [1,1,4,1,1] => ? = 1
[2,1,1,1,3] => 10111100 => [1,1,4,2] => [2,4,1,1] => ? = 1
[2,1,2,1,1,1] => 10110111 => [1,1,2,1,3] => [3,1,2,1,1] => ? = 1
[2,1,5] => 10110000 => [1,1,2,4] => [4,2,1,1] => ? = 1
[2,2,1,1,1,1] => 10101111 => [1,1,1,1,4] => [4,1,1,1,1] => ? = 1
[2,2,2,1,1] => 10101011 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1] => ? = 1
[2,2,4] => 10101000 => [1,1,1,1,1,3] => [3,1,1,1,1,1] => ? = 1
[2,3,1,1,1] => 10100111 => [1,1,1,2,3] => [3,2,1,1,1] => ? = 1
[2,5,1] => 10100001 => [1,1,1,4,1] => [1,4,1,1,1] => ? = 1
[3,2,1,1,1] => 10010111 => [1,2,1,1,3] => [3,1,1,2,1] => ? = 1
[5,2,1] => 10000101 => [1,4,1,1,1] => [1,1,1,4,1] => ? = 1
[6,1,1] => 10000011 => [1,5,2] => [2,5,1] => ? = 1
[1,1,1,1,1,1,1,1,1] => 111111111 => [9] => [9] => ? = 9
[1,1,1,1,1,1,2,1] => 111111101 => [7,1,1] => [1,1,7] => ? = 7
[1,1,1,1,1,1,3] => 111111100 => [7,2] => [2,7] => ? = 7
[1,1,1,1,1,2,1,1] => 111111011 => [6,1,2] => [2,1,6] => ? = 6
[1,1,1,1,1,2,2] => 111111010 => [6,1,1,1] => [1,1,1,6] => ? = 6
[1,1,1,1,1,3,1] => 111111001 => [6,2,1] => [1,2,6] => ? = 6
[1,1,1,1,1,4] => 111111000 => [6,3] => [3,6] => ? = 6
[1,1,1,1,2,1,1,1] => 111110111 => [5,1,3] => [3,1,5] => ? = 5
[1,1,1,1,2,1,2] => 111110110 => [5,1,2,1] => [1,2,1,5] => ? = 5
[1,1,1,1,2,2,1] => 111110101 => [5,1,1,1,1] => [1,1,1,1,5] => ? = 5
[1,1,1,1,2,3] => 111110100 => [5,1,1,2] => [2,1,1,5] => ? = 5
Description
The last part of an integer composition.
Matching statistic: St000011
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 2
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2
[1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 2
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 2
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 1
[1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,2,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ?
=> ? = 4
[1,1,1,2,1,3] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ?
=> ?
=> ? = 4
[1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,2,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ?
=> ?
=> ? = 4
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,3,3] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,4,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ?
=> ? = 4
[1,1,1,5,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,2,1,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ?
=> ?
=> ? = 3
[1,1,2,1,1,3] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,2,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ?
=> ?
=> ? = 3
[1,1,2,1,2,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 3
[1,1,2,1,3,1] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ?
=> ?
=> ? = 3
[1,1,2,1,4] => [1,0,1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ?
=> ?
=> ? = 3
[1,1,2,2,1,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 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: St000439
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 5 = 4 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 5 + 1
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5 + 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 4 + 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6 + 1
[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6 + 1
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 6 + 1
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 + 1
[1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 + 1
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ?
=> ? = 5 + 1
[1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 5 + 1
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 + 1
[1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5 + 1
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 + 1
[1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5 + 1
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4 + 1
[1,1,1,2,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 4 + 1
[1,1,1,2,1,3] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ?
=> ?
=> ? = 4 + 1
[1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[1,1,1,2,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ?
=> ?
=> ? = 4 + 1
[1,1,1,2,4] => [1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,1,0,0,0]
=> ?
=> ? = 4 + 1
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 4 + 1
[1,1,1,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,1,0,0]
=> ?
=> ? = 4 + 1
[1,1,1,3,3] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,0]
=> ?
=> ? = 4 + 1
[1,1,1,4,1,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,1,4,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,1,0]
=> ?
=> ? = 4 + 1
[1,1,1,5,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,1,6] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3 + 1
[1,1,2,1,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000745
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 90%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 90%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 1
[1,1] => [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 2
[2] => [1,1,0,0]
=> [1,2] => [[1,2]]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => [[1,2],[3]]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,2],[3],[4]]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,2,3],[4]]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,5],[2],[3],[4]]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,4],[2],[3],[5]]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,4,5],[2],[3]]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,3],[2],[4],[5]]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,3],[2,5],[4]]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[1,2],[3],[4],[5]]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[1,2],[3,4],[5]]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,5],[3,4]]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[1,2,3],[4],[5]]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,3],[4,5]]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,2,3,4],[5]]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5]]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [[1,5],[2],[3],[4],[6]]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [[1,5,6],[2],[3],[4]]
=> 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6]]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [[1,4],[2,6],[3],[5]]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [[1,4,5],[2],[3],[6]]
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [[1,4,5,6],[2],[3]]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [[1,3],[2],[4],[5],[6]]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [[1,3],[2,6],[4],[5]]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [[1,3,6],[2,5],[4]]
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [[1,3,4],[2],[5],[6]]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [[1,3,4],[2,6],[5]]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [[1,3,4,5],[2],[6]]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,3,4,5,6],[2]]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [[1,2],[3],[4],[5],[6]]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [[1,2],[3,6],[4],[5]]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [[1,2],[3,5],[4],[6]]
=> 1
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,3,2,1] => ?
=> ? = 4
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ? = 4
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,2,1] => ?
=> ? = 3
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,4,2,1] => ?
=> ? = 3
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,4,2,3,1] => ?
=> ? = 2
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => ?
=> ? = 2
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,2,3,1] => ?
=> ? = 2
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,2,3,1] => ?
=> ? = 2
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,2,3,4,1] => ?
=> ? = 2
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,1,2] => ?
=> ? = 1
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,1,2] => ?
=> ? = 1
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,1,2] => ?
=> ? = 1
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,1,2] => ?
=> ? = 1
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,1,2] => ?
=> ? = 1
[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,5,1,2] => ?
=> ? = 1
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [6,7,8,3,4,5,1,2] => ?
=> ? = 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,1,2,3] => ?
=> ? = 1
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,1,2,3] => ?
=> ? = 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4,5,1,2,3] => ?
=> ? = 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,1,2,3] => ?
=> ? = 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,1,2,3] => ?
=> ? = 1
[1,1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [9,7,8,6,5,4,3,2,1] => [[1,8],[2],[3],[4],[5],[6],[7],[9]]
=> ? = 7
[1,1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [9,8,6,7,5,4,3,2,1] => ?
=> ? = 6
[1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [9,6,7,8,5,4,3,2,1] => [[1,7,8],[2],[3],[4],[5],[6],[9]]
=> ? = 6
[1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [6,7,8,9,5,4,3,2,1] => ?
=> ? = 6
[1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [9,8,7,5,6,4,3,2,1] => [[1,6],[2],[3],[4],[5],[7],[8],[9]]
=> ? = 5
[1,1,1,1,2,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [8,9,7,5,6,4,3,2,1] => ?
=> ? = 5
[1,1,1,1,2,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [9,7,8,5,6,4,3,2,1] => ?
=> ? = 5
[1,1,1,1,2,3] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [7,8,9,5,6,4,3,2,1] => ?
=> ? = 5
[1,1,1,1,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [9,8,5,6,7,4,3,2,1] => ?
=> ? = 5
[1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [8,9,5,6,7,4,3,2,1] => ?
=> ? = 5
[1,1,1,1,4,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [9,5,6,7,8,4,3,2,1] => ?
=> ? = 5
[1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [5,6,7,8,9,4,3,2,1] => ?
=> ? = 5
[1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,4,5,3,2,1] => [[1,5],[2],[3],[4],[6],[7],[8],[9]]
=> ? = 4
[1,1,1,2,1,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 4
[1,1,1,2,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [9,7,8,6,4,5,3,2,1] => ?
=> ? = 4
[1,1,1,2,1,3] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? => ?
=> ? = 4
[1,1,1,2,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [9,8,6,7,4,5,3,2,1] => ?
=> ? = 4
[1,1,1,2,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [8,9,6,7,4,5,3,2,1] => ?
=> ? = 4
[1,1,1,2,3,1] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? => ?
=> ? = 4
[1,1,1,2,4] => [1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [6,7,8,9,4,5,3,2,1] => ?
=> ? = 4
[1,1,1,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [9,8,7,4,5,6,3,2,1] => [[1,5,6],[2],[3],[4],[7],[8],[9]]
=> ? = 4
[1,1,1,3,1,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 4
[1,1,1,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [9,7,8,4,5,6,3,2,1] => ?
=> ? = 4
[1,1,1,3,3] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [7,8,9,4,5,6,3,2,1] => ?
=> ? = 4
[1,1,1,4,1,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? => ?
=> ? = 4
[1,1,1,4,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? => ?
=> ? = 4
[1,1,1,5,1] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [9,4,5,6,7,8,3,2,1] => ?
=> ? = 4
[1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6],[7],[8],[9]]
=> ? = 3
[1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [8,9,7,6,5,3,4,2,1] => ?
=> ? = 3
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000759
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> []
=> 1
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [1]
=> 2
[2] => [1,1,0,0]
=> [1,1,0,0]
=> []
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,2,1]
=> 4
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,2,1]
=> 3
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? = 5
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? = 4
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 3
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> ? = 3
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 3
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> ? = 2
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> ? = 2
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 2
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2]
=> ? = 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? = 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> ? = 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 1
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1]
=> ? = 6
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> ? = 5
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1]
=> ? = 5
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1]
=> ? = 5
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 4
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,1]
=> ? = 4
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1]
=> ? = 4
[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1]
=> ? = 4
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1]
=> ? = 4
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1]
=> ? = 3
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1]
=> ? = 3
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,1]
=> ? = 3
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,1]
=> ? = 3
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> [6,4,2,1]
=> ? = 3
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1]
=> ? = 3
[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,1]
=> ? = 3
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> [6,5,2,1]
=> ? = 3
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1]
=> ? = 3
[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2,1]
=> ? = 3
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1]
=> ? = 2
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,1]
=> ? = 2
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [6,4,3,1]
=> ? = 2
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1]
=> ? = 2
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,1]
=> ? = 2
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,1,0,0]
=> [6,5,3,1]
=> ? = 2
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> [7,6,3,1]
=> ? = 2
[1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1]
=> ? = 2
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1]
=> ? = 2
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1]
=> ? = 2
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> [7,6,4,1]
=> ? = 2
[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1]
=> ? = 2
[1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1]
=> ? = 2
[2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2]
=> ? = 1
[2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2]
=> ? = 1
[2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2]
=> ? = 1
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2]
=> ? = 1
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000678
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 80%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1,0]
=> ? = 1
[1,1] => [1,1] => [1,0,1,0]
=> 2
[2] => [2] => [1,1,0,0]
=> 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3] => [3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 4
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 2
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 2
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
[1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,1,2,4] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3
[1,1,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 3
[1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,2,5] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,3,4] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[2,1,1,4] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 1
[2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[2,1,5] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[2,2,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[3,1,4] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[4,1,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,1,1,1,3] => [3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[1,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,1,2,2] => [2,2,1,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,1,3,1] => [1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,1,4] => [4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[1,1,1,1,2,1,1,1] => [1,1,1,2,1,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,2,1,2] => [2,1,2,1,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,2,2,1] => [1,2,2,1,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,2,3] => [3,2,1,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,3,1,1] => [1,1,3,1,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,3,2] => [2,3,1,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,4,1] => [1,4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,1,5] => [5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
[1,1,1,2,1,1,1,1] => [1,1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 4
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000475
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 90%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 90%
Values
[1] => [[1],[]]
=> [1]
=> []
=> 0 = 1 - 1
[1,1] => [[1,1],[]]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2] => [[2],[]]
=> [2]
=> []
=> 0 = 1 - 1
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,2] => [[2,1],[]]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,1] => [[2,2],[1]]
=> [2,2]
=> [2]
=> 0 = 1 - 1
[3] => [[3],[]]
=> [3]
=> []
=> 0 = 1 - 1
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3] => [[3,1],[]]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 0 = 1 - 1
[2,2] => [[3,2],[1]]
=> [3,2]
=> [2]
=> 0 = 1 - 1
[3,1] => [[3,3],[2]]
=> [3,3]
=> [3]
=> 0 = 1 - 1
[4] => [[4],[]]
=> [4]
=> []
=> 0 = 1 - 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
[1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 1 = 2 - 1
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> [3,1]
=> 1 = 2 - 1
[1,4] => [[4,1],[]]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> 0 = 1 - 1
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> [2,2]
=> 0 = 1 - 1
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> [3,2]
=> 0 = 1 - 1
[2,3] => [[4,2],[1]]
=> [4,2]
=> [2]
=> 0 = 1 - 1
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3]
=> 0 = 1 - 1
[3,2] => [[4,3],[2]]
=> [4,3]
=> [3]
=> 0 = 1 - 1
[4,1] => [[4,4],[3]]
=> [4,4]
=> [4]
=> 0 = 1 - 1
[5] => [[5],[]]
=> [5]
=> []
=> 0 = 1 - 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 3 = 4 - 1
[1,1,1,3] => [[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 2 = 3 - 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[1,1,4] => [[4,1,1],[]]
=> [4,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 1 = 2 - 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> [2,2,1]
=> 1 = 2 - 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> [3,2,1]
=> 1 = 2 - 1
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [3,3,1]
=> 1 = 2 - 1
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> [3,1]
=> 1 = 2 - 1
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> [4,1]
=> 1 = 2 - 1
[1,5] => [[5,1],[]]
=> [5,1]
=> [1]
=> 1 = 2 - 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> 0 = 1 - 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> [2,2,2]
=> 0 = 1 - 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [3,2,2]
=> 0 = 1 - 1
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? = 5 - 1
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? = 4 - 1
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? = 4 - 1
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? = 4 - 1
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? = 3 - 1
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? = 3 - 1
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? = 3 - 1
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? = 3 - 1
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? = 3 - 1
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? = 3 - 1
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? = 2 - 1
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? = 2 - 1
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? = 2 - 1
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? = 2 - 1
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? = 2 - 1
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? = 2 - 1
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
=> [4,4,4,4,1]
=> [4,4,4,1]
=> ? = 2 - 1
[1,5,1,1] => [[5,5,5,1],[4,4]]
=> ?
=> ?
=> ? = 2 - 1
[1,5,2] => [[6,5,1],[4]]
=> ?
=> ?
=> ? = 2 - 1
[1,6,1] => [[6,6,1],[5]]
=> ?
=> ?
=> ? = 2 - 1
[2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
=> ?
=> ?
=> ? = 1 - 1
[2,1,1,4] => [[5,2,2,2],[1,1,1]]
=> ?
=> ?
=> ? = 1 - 1
[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> ? = 1 - 1
[2,1,4,1] => [[5,5,2,2],[4,1,1]]
=> ?
=> ?
=> ? = 1 - 1
[2,1,5] => [[6,2,2],[1,1]]
=> ?
=> ?
=> ? = 1 - 1
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
=> [3,3,3,3,3,2]
=> [3,3,3,3,2]
=> ? = 1 - 1
[2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]]
=> [4,4,4,3,2]
=> [4,4,3,2]
=> ? = 1 - 1
[2,2,4] => [[6,3,2],[2,1]]
=> ?
=> ?
=> ? = 1 - 1
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
=> [4,4,4,4,2]
=> [4,4,4,2]
=> ? = 1 - 1
[2,5,1] => [[6,6,2],[5,1]]
=> ?
=> ?
=> ? = 1 - 1
[2,6] => [[7,2],[1]]
=> ?
=> ?
=> ? = 1 - 1
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> ? = 1 - 1
[3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
=> ?
=> ?
=> ? = 1 - 1
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]]
=> [4,4,3,3,3]
=> [4,3,3,3]
=> ? = 1 - 1
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
=> [4,4,4,3,3]
=> ?
=> ? = 1 - 1
[3,1,4] => [[6,3,3],[2,2]]
=> ?
=> ?
=> ? = 1 - 1
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
=> [4,4,4,4,3]
=> ?
=> ? = 1 - 1
[3,3,1,1] => [[5,5,5,3],[4,4,2]]
=> [5,5,5,3]
=> ?
=> ? = 1 - 1
[3,5] => [[7,3],[2]]
=> ?
=> ?
=> ? = 1 - 1
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> [4,4,4,4,4]
=> [4,4,4,4]
=> ? = 1 - 1
[4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> [5,5,4,4]
=> ?
=> ? = 1 - 1
[4,2,1,1] => [[5,5,5,4],[4,4,3]]
=> [5,5,5,4]
=> ?
=> ? = 1 - 1
[5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> [5,5,5,5]
=> [5,5,5]
=> ? = 1 - 1
[5,1,2] => [[6,5,5],[4,4]]
=> ?
=> ?
=> ? = 1 - 1
[5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ?
=> ? = 1 - 1
[5,3] => [[7,5],[4]]
=> ?
=> ?
=> ? = 1 - 1
[6,2] => [[7,6],[5]]
=> ?
=> ?
=> ? = 1 - 1
[1,1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1,1],[1]]
=> ?
=> ?
=> ? = 7 - 1
[1,1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1,1],[1,1]]
=> ?
=> ?
=> ? = 6 - 1
[1,1,1,1,1,2,2] => [[3,2,1,1,1,1,1],[1]]
=> ?
=> ?
=> ? = 6 - 1
Description
The number of parts equal to 1 in a partition.
The following 93 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St001176The size of a partition minus its first part. St000700The protection number of an ordered tree. St001733The number of weak left to right maxima of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000993The multiplicity of the largest part of an integer partition. St000617The number of global maxima of a Dyck path. St000160The multiplicity of the smallest part of a partition. St000738The first entry in the last row of a standard tableau. St000237The number of small exceedances. St000054The first entry of the permutation. St000505The biggest entry in the block containing the 1. St000883The number of longest increasing subsequences of a permutation. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000007The number of saliances of the permutation. St000546The number of global descents of a permutation. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001691The number of kings in a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001829The common independence number of a graph. St000234The number of global ascents of a permutation. St000363The number of minimal vertex covers of a graph. St000287The number of connected components of a graph. St000553The number of blocks of a graph. St000917The open packing number of a graph. St001672The restrained domination number of a graph. St001316The domatic number of a graph. St001828The Euler characteristic of a graph. St000914The sum of the values of the Möbius function of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000286The number of connected components of the complement of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000989The number of final rises of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000740The last entry of a permutation. St000654The first descent of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000181The number of connected components of the Hasse diagram for the poset. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001461The number of topologically connected components of the chord diagram of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000335The difference of lower and upper interactions. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000051The size of the left subtree of a binary tree. St000090The variation of a composition. St000133The "bounce" of a permutation. St000258The burning number of a graph. St000699The toughness times the least common multiple of 1,. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000061The number of nodes on the left branch of a binary tree. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000456The monochromatic index of a connected graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000338The number of pixed points of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral.
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