Your data matches 57 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
(load all 8 compositions to match this statistic)
Mapping
Mp00109: Permutations —descent word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0
[2,1] => 1 => 1
[1,2,3] => 00 => 0
[1,3,2] => 01 => 0
[2,1,3] => 10 => 1
[2,3,1] => 01 => 0
[3,1,2] => 10 => 1
[3,2,1] => 11 => 2
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 0
[1,3,2,4] => 010 => 0
[1,3,4,2] => 001 => 0
[1,4,2,3] => 010 => 0
[1,4,3,2] => 011 => 0
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 1
[2,3,1,4] => 010 => 0
[2,3,4,1] => 001 => 0
[2,4,1,3] => 010 => 0
[2,4,3,1] => 011 => 0
[3,1,2,4] => 100 => 1
[3,1,4,2] => 101 => 1
[3,2,1,4] => 110 => 2
[3,2,4,1] => 101 => 1
[3,4,1,2] => 010 => 0
[3,4,2,1] => 011 => 0
[4,1,2,3] => 100 => 1
[4,1,3,2] => 101 => 1
[4,2,1,3] => 110 => 2
[4,2,3,1] => 101 => 1
[4,3,1,2] => 110 => 2
[4,3,2,1] => 111 => 3
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 0
[1,2,4,3,5] => 0010 => 0
[1,2,4,5,3] => 0001 => 0
[1,2,5,3,4] => 0010 => 0
[1,2,5,4,3] => 0011 => 0
[1,3,2,4,5] => 0100 => 0
[1,3,2,5,4] => 0101 => 0
[1,3,4,2,5] => 0010 => 0
[1,3,4,5,2] => 0001 => 0
[1,3,5,2,4] => 0010 => 0
[1,3,5,4,2] => 0011 => 0
[1,4,2,3,5] => 0100 => 0
[1,4,2,5,3] => 0101 => 0
[1,4,3,2,5] => 0110 => 0
[1,4,3,5,2] => 0101 => 0
[1,4,5,2,3] => 0010 => 0
[1,4,5,3,2] => 0011 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 93%●distinct values known / distinct values provided: 91%
Values
[1,2] => [2] => 10 => 01 => 2 = 0 + 2
[2,1] => [1,1] => 11 => 00 => 3 = 1 + 2
[1,2,3] => [3] => 100 => 011 => 2 = 0 + 2
[1,3,2] => [2,1] => 101 => 010 => 2 = 0 + 2
[2,1,3] => [1,2] => 110 => 001 => 3 = 1 + 2
[2,3,1] => [2,1] => 101 => 010 => 2 = 0 + 2
[3,1,2] => [1,2] => 110 => 001 => 3 = 1 + 2
[3,2,1] => [1,1,1] => 111 => 000 => 4 = 2 + 2
[1,2,3,4] => [4] => 1000 => 0111 => 2 = 0 + 2
[1,2,4,3] => [3,1] => 1001 => 0110 => 2 = 0 + 2
[1,3,2,4] => [2,2] => 1010 => 0101 => 2 = 0 + 2
[1,3,4,2] => [3,1] => 1001 => 0110 => 2 = 0 + 2
[1,4,2,3] => [2,2] => 1010 => 0101 => 2 = 0 + 2
[1,4,3,2] => [2,1,1] => 1011 => 0100 => 2 = 0 + 2
[2,1,3,4] => [1,3] => 1100 => 0011 => 3 = 1 + 2
[2,1,4,3] => [1,2,1] => 1101 => 0010 => 3 = 1 + 2
[2,3,1,4] => [2,2] => 1010 => 0101 => 2 = 0 + 2
[2,3,4,1] => [3,1] => 1001 => 0110 => 2 = 0 + 2
[2,4,1,3] => [2,2] => 1010 => 0101 => 2 = 0 + 2
[2,4,3,1] => [2,1,1] => 1011 => 0100 => 2 = 0 + 2
[3,1,2,4] => [1,3] => 1100 => 0011 => 3 = 1 + 2
[3,1,4,2] => [1,2,1] => 1101 => 0010 => 3 = 1 + 2
[3,2,1,4] => [1,1,2] => 1110 => 0001 => 4 = 2 + 2
[3,2,4,1] => [1,2,1] => 1101 => 0010 => 3 = 1 + 2
[3,4,1,2] => [2,2] => 1010 => 0101 => 2 = 0 + 2
[3,4,2,1] => [2,1,1] => 1011 => 0100 => 2 = 0 + 2
[4,1,2,3] => [1,3] => 1100 => 0011 => 3 = 1 + 2
[4,1,3,2] => [1,2,1] => 1101 => 0010 => 3 = 1 + 2
[4,2,1,3] => [1,1,2] => 1110 => 0001 => 4 = 2 + 2
[4,2,3,1] => [1,2,1] => 1101 => 0010 => 3 = 1 + 2
[4,3,1,2] => [1,1,2] => 1110 => 0001 => 4 = 2 + 2
[4,3,2,1] => [1,1,1,1] => 1111 => 0000 => 5 = 3 + 2
[1,2,3,4,5] => [5] => 10000 => 01111 => 2 = 0 + 2
[1,2,3,5,4] => [4,1] => 10001 => 01110 => 2 = 0 + 2
[1,2,4,3,5] => [3,2] => 10010 => 01101 => 2 = 0 + 2
[1,2,4,5,3] => [4,1] => 10001 => 01110 => 2 = 0 + 2
[1,2,5,3,4] => [3,2] => 10010 => 01101 => 2 = 0 + 2
[1,2,5,4,3] => [3,1,1] => 10011 => 01100 => 2 = 0 + 2
[1,3,2,4,5] => [2,3] => 10100 => 01011 => 2 = 0 + 2
[1,3,2,5,4] => [2,2,1] => 10101 => 01010 => 2 = 0 + 2
[1,3,4,2,5] => [3,2] => 10010 => 01101 => 2 = 0 + 2
[1,3,4,5,2] => [4,1] => 10001 => 01110 => 2 = 0 + 2
[1,3,5,2,4] => [3,2] => 10010 => 01101 => 2 = 0 + 2
[1,3,5,4,2] => [3,1,1] => 10011 => 01100 => 2 = 0 + 2
[1,4,2,3,5] => [2,3] => 10100 => 01011 => 2 = 0 + 2
[1,4,2,5,3] => [2,2,1] => 10101 => 01010 => 2 = 0 + 2
[1,4,3,2,5] => [2,1,2] => 10110 => 01001 => 2 = 0 + 2
[1,4,3,5,2] => [2,2,1] => 10101 => 01010 => 2 = 0 + 2
[1,4,5,2,3] => [3,2] => 10010 => 01101 => 2 = 0 + 2
[1,4,5,3,2] => [3,1,1] => 10011 => 01100 => 2 = 0 + 2
[7,6,5,3,2,4,8,1] => ? => ? => ? => ? = 4 + 2
[8,5,4,6,3,2,1,7] => ? => ? => ? => ? = 2 + 2
[5,6,4,3,2,7,1,8] => ? => ? => ? => ? = 0 + 2
[7,5,4,3,1,2,6,8] => ? => ? => ? => ? = 4 + 2
[4,3,5,6,1,2,7,8] => ? => ? => ? => ? = 1 + 2
[5,6,2,1,3,4,7,8] => ? => ? => ? => ? = 0 + 2
[4,3,5,1,2,6,7,8] => ? => ? => ? => ? = 1 + 2
[4,5,2,1,3,6,7,8] => ? => ? => ? => ? = 0 + 2
[4,5,6,1,2,3,8,7,10,9] => [3,4,2,1] => 1001000101 => 0110111010 => ? = 0 + 2
[2,1,5,6,3,4,8,7,10,9] => [1,3,3,2,1] => 1100100101 => 0011011010 => ? = 1 + 2
[8,9,5,6,3,4,10,1,2,7] => [2,2,3,3] => 1010100100 => 0101011011 => ? = 0 + 2
[5,6,7,8,1,2,3,4,10,9] => [4,5,1] => 1000100001 => 0111011110 => ? = 0 + 2
[4,6,7,1,8,2,3,5,10,9] => [3,2,4,1] => 1001010001 => 0110101110 => ? = 0 + 2
[3,6,1,7,8,2,4,5,10,9] => [2,3,4,1] => 1010010001 => 0101101110 => ? = 0 + 2
[2,1,6,7,8,3,4,5,10,9] => [1,4,4,1] => 1100010001 => 0011101110 => ? = 1 + 2
[2,1,5,7,3,8,4,6,10,9] => [1,3,2,3,1] => 1100101001 => 0011010110 => ? = 1 + 2
[4,5,7,1,2,8,3,6,10,9] => [3,3,3,1] => 1001001001 => 0110110110 => ? = 0 + 2
[3,4,1,2,7,8,5,6,10,9] => [2,4,3,1] => 1010001001 => 0101110110 => ? = 0 + 2
[2,1,7,8,9,10,3,4,5,6] => [1,5,4] => 1100001000 => 0011110111 => ? = 1 + 2
[3,7,1,8,9,10,2,4,5,6] => [2,4,4] => 1010001000 => 0101110111 => ? = 0 + 2
[4,7,8,1,9,10,2,3,5,6] => [3,3,4] => 1001001000 => 0110110111 => ? = 0 + 2
[5,7,8,9,1,10,2,3,4,6] => [4,2,4] => 1000101000 => 0111010111 => ? = 0 + 2
[6,7,8,9,10,1,2,3,4,5] => [5,5] => 1000010000 => 0111101111 => ? = 0 + 2
[5,6,8,9,1,2,10,3,4,7] => [4,3,3] => 1000100100 => 0111011011 => ? = 0 + 2
[4,6,8,1,9,2,10,3,5,7] => [3,2,2,3] => 1001010100 => 0110101011 => ? = 0 + 2
[3,6,1,8,9,2,10,4,5,7] => [2,3,2,3] => 1010010100 => 0101101011 => ? = 0 + 2
[2,1,6,8,9,3,10,4,5,7] => [1,4,2,3] => 1100010100 => 0011101011 => ? = 1 + 2
[2,1,5,8,3,9,10,4,6,7] => [1,3,3,3] => 1100100100 => 0011011011 => ? = 1 + 2
[3,5,1,8,2,9,10,4,6,7] => [2,2,3,3] => 1010100100 => 0101011011 => ? = 0 + 2
[4,5,8,1,2,9,10,3,6,7] => [3,4,3] => 1001000100 => 0110111011 => ? = 0 + 2
[3,4,1,2,8,9,10,5,6,7] => [2,5,3] => 1010000100 => 0101111011 => ? = 0 + 2
[2,1,4,3,8,9,10,5,6,7] => [1,2,4,3] => 1101000100 => 0010111011 => ? = 1 + 2
[3,4,1,2,7,9,5,10,6,8] => [2,4,2,2] => 1010001010 => 0101110101 => ? = 0 + 2
[4,5,7,1,2,9,3,10,6,8] => [3,3,2,2] => 1001001010 => 0110110101 => ? = 0 + 2
[2,1,6,7,9,3,4,10,5,8] => [1,4,3,2] => 1100010010 => 0011101101 => ? = 1 + 2
[3,6,1,7,9,2,4,10,5,8] => [2,3,3,2] => 1010010010 => 0101101101 => ? = 0 + 2
[4,6,7,1,9,2,3,10,5,8] => [3,2,3,2] => 1001010010 => 0110101101 => ? = 0 + 2
[5,6,7,9,1,2,3,10,4,8] => [4,4,2] => 1000100010 => 0111011101 => ? = 0 + 2
[2,1,5,6,3,4,9,10,7,8] => [1,3,4,2] => 1100100010 => 0011011101 => ? = 1 + 2
[3,5,1,6,2,4,9,10,7,8] => [2,2,4,2] => 1010100010 => 0101011101 => ? = 0 + 2
[4,5,6,1,2,3,9,10,7,8] => [3,5,2] => 1001000010 => 0110111101 => ? = 0 + 2
[3,4,1,2,6,5,9,10,7,8] => [2,3,3,2] => 1010010010 => 0101101101 => ? = 0 + 2
[3,4,2,1,6,5,8,7,10,9] => [2,1,2,2,2,1] => 1011010101 => 0100101010 => ? = 0 + 2
[3,5,2,6,4,1,8,7,10,9] => [2,2,1,2,2,1] => 1010110101 => 0101001010 => ? = 0 + 2
[3,5,2,7,4,8,6,1,10,9] => [2,2,2,1,2,1] => 1010101101 => 0101010010 => ? = 0 + 2
[3,5,2,7,4,9,6,10,8,1] => [2,2,2,2,1,1] => 1010101011 => 0101010100 => ? = 0 + 2
[2,1,5,6,4,3,8,7,10,9] => [1,3,1,2,2,1] => 1100110101 => 0011001010 => ? = 1 + 2
[4,5,6,3,2,1,8,7,10,9] => [3,1,1,2,2,1] => 1001110101 => 0110001010 => ? = 0 + 2
[4,5,7,3,2,8,6,1,10,9] => [3,1,2,1,2,1] => 1001101101 => 0110010010 => ? = 0 + 2
[4,5,7,3,2,9,6,10,8,1] => [3,1,2,2,1,1] => 1001101011 => 0110010100 => ? = 0 + 2
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 6 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
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: 92%●distinct values known / distinct values provided: 91%
Values
[1,2] => [2] => 10 => [1,1] => 1 = 0 + 1
[2,1] => [1,1] => 11 => [2] => 2 = 1 + 1
[1,2,3] => [3] => 100 => [1,2] => 1 = 0 + 1
[1,3,2] => [2,1] => 101 => [1,1,1] => 1 = 0 + 1
[2,1,3] => [1,2] => 110 => [2,1] => 2 = 1 + 1
[2,3,1] => [2,1] => 101 => [1,1,1] => 1 = 0 + 1
[3,1,2] => [1,2] => 110 => [2,1] => 2 = 1 + 1
[3,2,1] => [1,1,1] => 111 => [3] => 3 = 2 + 1
[1,2,3,4] => [4] => 1000 => [1,3] => 1 = 0 + 1
[1,2,4,3] => [3,1] => 1001 => [1,2,1] => 1 = 0 + 1
[1,3,2,4] => [2,2] => 1010 => [1,1,1,1] => 1 = 0 + 1
[1,3,4,2] => [3,1] => 1001 => [1,2,1] => 1 = 0 + 1
[1,4,2,3] => [2,2] => 1010 => [1,1,1,1] => 1 = 0 + 1
[1,4,3,2] => [2,1,1] => 1011 => [1,1,2] => 1 = 0 + 1
[2,1,3,4] => [1,3] => 1100 => [2,2] => 2 = 1 + 1
[2,1,4,3] => [1,2,1] => 1101 => [2,1,1] => 2 = 1 + 1
[2,3,1,4] => [2,2] => 1010 => [1,1,1,1] => 1 = 0 + 1
[2,3,4,1] => [3,1] => 1001 => [1,2,1] => 1 = 0 + 1
[2,4,1,3] => [2,2] => 1010 => [1,1,1,1] => 1 = 0 + 1
[2,4,3,1] => [2,1,1] => 1011 => [1,1,2] => 1 = 0 + 1
[3,1,2,4] => [1,3] => 1100 => [2,2] => 2 = 1 + 1
[3,1,4,2] => [1,2,1] => 1101 => [2,1,1] => 2 = 1 + 1
[3,2,1,4] => [1,1,2] => 1110 => [3,1] => 3 = 2 + 1
[3,2,4,1] => [1,2,1] => 1101 => [2,1,1] => 2 = 1 + 1
[3,4,1,2] => [2,2] => 1010 => [1,1,1,1] => 1 = 0 + 1
[3,4,2,1] => [2,1,1] => 1011 => [1,1,2] => 1 = 0 + 1
[4,1,2,3] => [1,3] => 1100 => [2,2] => 2 = 1 + 1
[4,1,3,2] => [1,2,1] => 1101 => [2,1,1] => 2 = 1 + 1
[4,2,1,3] => [1,1,2] => 1110 => [3,1] => 3 = 2 + 1
[4,2,3,1] => [1,2,1] => 1101 => [2,1,1] => 2 = 1 + 1
[4,3,1,2] => [1,1,2] => 1110 => [3,1] => 3 = 2 + 1
[4,3,2,1] => [1,1,1,1] => 1111 => [4] => 4 = 3 + 1
[1,2,3,4,5] => [5] => 10000 => [1,4] => 1 = 0 + 1
[1,2,3,5,4] => [4,1] => 10001 => [1,3,1] => 1 = 0 + 1
[1,2,4,3,5] => [3,2] => 10010 => [1,2,1,1] => 1 = 0 + 1
[1,2,4,5,3] => [4,1] => 10001 => [1,3,1] => 1 = 0 + 1
[1,2,5,3,4] => [3,2] => 10010 => [1,2,1,1] => 1 = 0 + 1
[1,2,5,4,3] => [3,1,1] => 10011 => [1,2,2] => 1 = 0 + 1
[1,3,2,4,5] => [2,3] => 10100 => [1,1,1,2] => 1 = 0 + 1
[1,3,2,5,4] => [2,2,1] => 10101 => [1,1,1,1,1] => 1 = 0 + 1
[1,3,4,2,5] => [3,2] => 10010 => [1,2,1,1] => 1 = 0 + 1
[1,3,4,5,2] => [4,1] => 10001 => [1,3,1] => 1 = 0 + 1
[1,3,5,2,4] => [3,2] => 10010 => [1,2,1,1] => 1 = 0 + 1
[1,3,5,4,2] => [3,1,1] => 10011 => [1,2,2] => 1 = 0 + 1
[1,4,2,3,5] => [2,3] => 10100 => [1,1,1,2] => 1 = 0 + 1
[1,4,2,5,3] => [2,2,1] => 10101 => [1,1,1,1,1] => 1 = 0 + 1
[1,4,3,2,5] => [2,1,2] => 10110 => [1,1,2,1] => 1 = 0 + 1
[1,4,3,5,2] => [2,2,1] => 10101 => [1,1,1,1,1] => 1 = 0 + 1
[1,4,5,2,3] => [3,2] => 10010 => [1,2,1,1] => 1 = 0 + 1
[1,4,5,3,2] => [3,1,1] => 10011 => [1,2,2] => 1 = 0 + 1
[7,6,5,3,2,4,8,1] => ? => ? => ? => ? = 4 + 1
[8,5,4,6,3,2,1,7] => ? => ? => ? => ? = 2 + 1
[5,6,4,3,2,7,1,8] => ? => ? => ? => ? = 0 + 1
[7,5,4,3,1,2,6,8] => ? => ? => ? => ? = 4 + 1
[4,3,5,6,1,2,7,8] => ? => ? => ? => ? = 1 + 1
[5,6,2,1,3,4,7,8] => ? => ? => ? => ? = 0 + 1
[4,3,5,1,2,6,7,8] => ? => ? => ? => ? = 1 + 1
[4,5,2,1,3,6,7,8] => ? => ? => ? => ? = 0 + 1
[6,3,2,5,4,1,8,7,10,9] => [1,1,2,1,2,2,1] => 1110110101 => [3,1,2,1,1,1,1] => ? = 2 + 1
[8,3,2,5,4,7,6,1,10,9] => [1,1,2,2,1,2,1] => 1110101101 => [3,1,1,1,2,1,1] => ? = 2 + 1
[10,3,2,5,4,7,6,9,8,1] => [1,1,2,2,2,1,1] => 1110101011 => [3,1,1,1,1,1,2] => ? = 2 + 1
[4,5,6,1,2,3,8,7,10,9] => [3,4,2,1] => 1001000101 => ? => ? = 0 + 1
[2,1,6,5,4,3,8,7,10,9] => [1,2,1,1,2,2,1] => 1101110101 => ? => ? = 1 + 1
[8,5,4,3,2,7,6,1,10,9] => [1,1,1,1,2,1,2,1] => 1111101101 => ? => ? = 4 + 1
[10,5,4,3,2,7,6,9,8,1] => [1,1,1,1,2,2,1,1] => 1111101011 => ? => ? = 4 + 1
[2,1,8,5,4,7,6,3,10,9] => [1,2,1,2,1,2,1] => 1101101101 => ? => ? = 1 + 1
[8,7,4,3,6,5,2,1,10,9] => [1,1,1,2,1,1,2,1] => 1111011101 => [4,1,3,1,1] => ? = 3 + 1
[10,7,4,3,6,5,2,9,8,1] => [1,1,1,2,1,2,1,1] => 1111011011 => [4,1,2,1,2] => ? = 3 + 1
[2,1,10,5,4,7,6,9,8,3] => [1,2,1,2,2,1,1] => 1101101011 => ? => ? = 1 + 1
[10,9,4,3,6,5,8,7,2,1] => [1,1,1,2,2,1,1,1] => 1111010111 => [4,1,1,1,3] => ? = 3 + 1
[8,9,5,6,3,4,10,1,2,7] => [2,2,3,3] => 1010100100 => [1,1,1,1,1,2,1,2] => ? = 0 + 1
[5,6,7,8,1,2,3,4,10,9] => [4,5,1] => 1000100001 => ? => ? = 0 + 1
[4,6,7,1,8,2,3,5,10,9] => [3,2,4,1] => 1001010001 => ? => ? = 0 + 1
[2,1,6,7,8,3,4,5,10,9] => [1,4,4,1] => 1100010001 => ? => ? = 1 + 1
[2,1,5,7,3,8,4,6,10,9] => [1,3,2,3,1] => 1100101001 => [2,2,1,1,1,2,1] => ? = 1 + 1
[4,5,7,1,2,8,3,6,10,9] => [3,3,3,1] => 1001001001 => ? => ? = 0 + 1
[3,4,1,2,7,8,5,6,10,9] => [2,4,3,1] => 1010001001 => ? => ? = 0 + 1
[2,1,4,3,8,7,6,5,10,9] => [1,2,2,1,1,2,1] => 1101011101 => ? => ? = 1 + 1
[4,3,2,1,8,7,6,5,10,9] => [1,1,1,2,1,1,2,1] => 1111011101 => [4,1,3,1,1] => ? = 3 + 1
[8,3,2,7,6,5,4,1,10,9] => [1,1,2,1,1,1,2,1] => 1110111101 => [3,1,4,1,1] => ? = 2 + 1
[10,3,2,7,6,5,4,9,8,1] => [1,1,2,1,1,2,1,1] => 1110111011 => [3,1,3,1,2] => ? = 2 + 1
[2,1,8,7,6,5,4,3,10,9] => [1,2,1,1,1,1,2,1] => 1101111101 => [2,1,5,1,1] => ? = 1 + 1
[10,7,6,5,4,3,2,9,8,1] => [1,1,1,1,1,1,2,1,1] => 1111111011 => [7,1,2] => ? = 6 + 1
[2,1,10,7,6,5,4,9,8,3] => [1,2,1,1,1,2,1,1] => 1101111011 => [2,1,4,1,2] => ? = 1 + 1
[10,9,6,5,4,3,8,7,2,1] => [1,1,1,1,1,2,1,1,1] => 1111110111 => ? => ? = 5 + 1
[2,1,4,3,10,7,6,9,8,5] => [1,2,2,1,2,1,1] => 1101011011 => ? => ? = 1 + 1
[4,3,2,1,10,7,6,9,8,5] => [1,1,1,2,1,2,1,1] => 1111011011 => [4,1,2,1,2] => ? = 3 + 1
[10,3,2,9,6,5,8,7,4,1] => [1,1,2,1,2,1,1,1] => 1110110111 => ? => ? = 2 + 1
[2,1,10,9,6,5,8,7,4,3] => [1,2,1,1,2,1,1,1] => 1101110111 => [2,1,3,1,3] => ? = 1 + 1
[10,9,8,5,4,7,6,3,2,1] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? => ? = 4 + 1
[3,7,1,8,9,10,2,4,5,6] => [2,4,4] => 1010001000 => ? => ? = 0 + 1
[4,7,8,1,9,10,2,3,5,6] => [3,3,4] => 1001001000 => ? => ? = 0 + 1
[5,7,8,9,1,10,2,3,4,6] => [4,2,4] => 1000101000 => ? => ? = 0 + 1
[6,7,8,9,10,1,2,3,4,5] => [5,5] => 1000010000 => ? => ? = 0 + 1
[5,6,8,9,1,2,10,3,4,7] => [4,3,3] => 1000100100 => ? => ? = 0 + 1
[2,1,6,8,9,3,10,4,5,7] => [1,4,2,3] => 1100010100 => ? => ? = 1 + 1
[2,1,5,8,3,9,10,4,6,7] => [1,3,3,3] => 1100100100 => ? => ? = 1 + 1
[3,5,1,8,2,9,10,4,6,7] => [2,2,3,3] => 1010100100 => [1,1,1,1,1,2,1,2] => ? = 0 + 1
[4,5,8,1,2,9,10,3,6,7] => [3,4,3] => 1001000100 => ? => ? = 0 + 1
[7,3,2,8,9,10,1,4,5,6] => [1,1,4,4] => 1110001000 => ? => ? = 2 + 1
Description
The first part of an integer composition.
Matching statistic: St000745
(load all 5 compositions to match this statistic)
Mapping
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 91%
Values
[1,2] => [[1,2]]
 => 1 = 0 + 1
[2,1] => [[1],[2]]
 => 2 = 1 + 1
[1,2,3] => [[1,2,3]]
 => 1 = 0 + 1
[1,3,2] => [[1,2],[3]]
 => 1 = 0 + 1
[2,1,3] => [[1,3],[2]]
 => 2 = 1 + 1
[2,3,1] => [[1,2],[3]]
 => 1 = 0 + 1
[3,1,2] => [[1,3],[2]]
 => 2 = 1 + 1
[3,2,1] => [[1],[2],[3]]
 => 3 = 2 + 1
[1,2,3,4] => [[1,2,3,4]]
 => 1 = 0 + 1
[1,2,4,3] => [[1,2,3],[4]]
 => 1 = 0 + 1
[1,3,2,4] => [[1,2,4],[3]]
 => 1 = 0 + 1
[1,3,4,2] => [[1,2,3],[4]]
 => 1 = 0 + 1
[1,4,2,3] => [[1,2,4],[3]]
 => 1 = 0 + 1
[1,4,3,2] => [[1,2],[3],[4]]
 => 1 = 0 + 1
[2,1,3,4] => [[1,3,4],[2]]
 => 2 = 1 + 1
[2,1,4,3] => [[1,3],[2,4]]
 => 2 = 1 + 1
[2,3,1,4] => [[1,2,4],[3]]
 => 1 = 0 + 1
[2,3,4,1] => [[1,2,3],[4]]
 => 1 = 0 + 1
[2,4,1,3] => [[1,2],[3,4]]
 => 1 = 0 + 1
[2,4,3,1] => [[1,2],[3],[4]]
 => 1 = 0 + 1
[3,1,2,4] => [[1,3,4],[2]]
 => 2 = 1 + 1
[3,1,4,2] => [[1,3],[2,4]]
 => 2 = 1 + 1
[3,2,1,4] => [[1,4],[2],[3]]
 => 3 = 2 + 1
[3,2,4,1] => [[1,3],[2],[4]]
 => 2 = 1 + 1
[3,4,1,2] => [[1,2],[3,4]]
 => 1 = 0 + 1
[3,4,2,1] => [[1,2],[3],[4]]
 => 1 = 0 + 1
[4,1,2,3] => [[1,3,4],[2]]
 => 2 = 1 + 1
[4,1,3,2] => [[1,3],[2],[4]]
 => 2 = 1 + 1
[4,2,1,3] => [[1,4],[2],[3]]
 => 3 = 2 + 1
[4,2,3,1] => [[1,3],[2],[4]]
 => 2 = 1 + 1
[4,3,1,2] => [[1,4],[2],[3]]
 => 3 = 2 + 1
[4,3,2,1] => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[1,2,3,4,5] => [[1,2,3,4,5]]
 => 1 = 0 + 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
 => 1 = 0 + 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => 1 = 0 + 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
 => 1 = 0 + 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
 => 1 = 0 + 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 1 = 0 + 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => 1 = 0 + 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
 => 1 = 0 + 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
 => 1 = 0 + 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
 => 1 = 0 + 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
 => 1 = 0 + 1
[1,3,5,4,2] => [[1,2,3],[4],[5]]
 => 1 = 0 + 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
 => 1 = 0 + 1
[1,4,2,5,3] => [[1,2,4],[3,5]]
 => 1 = 0 + 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
 => 1 = 0 + 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
 => 1 = 0 + 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => 1 = 0 + 1
[1,4,5,3,2] => [[1,2,3],[4],[5]]
 => 1 = 0 + 1
[8,5,4,6,7,3,2,1] => ?
 => ? = 2 + 1
[8,6,4,5,3,7,2,1] => ?
 => ? = 2 + 1
[4,5,6,3,7,8,2,1] => ?
 => ? = 0 + 1
[8,7,6,4,2,3,5,1] => ?
 => ? = 4 + 1
[8,6,7,4,2,3,5,1] => ?
 => ? = 1 + 1
[8,5,6,4,2,3,7,1] => ?
 => ? = 1 + 1
[8,6,4,5,2,3,7,1] => ?
 => ? = 2 + 1
[8,5,2,3,4,6,7,1] => ?
 => ? = 2 + 1
[5,6,7,3,4,2,8,1] => ?
 => ? = 0 + 1
[6,4,5,7,2,3,8,1] => ?
 => ? = 1 + 1
[7,4,2,3,5,6,8,1] => ?
 => ? = 2 + 1
[6,5,7,4,3,8,1,2] => ?
 => ? = 1 + 1
[5,4,6,7,3,8,1,2] => ?
 => ? = 1 + 1
[7,5,6,3,4,8,1,2] => ?
 => ? = 1 + 1
[5,6,7,3,4,8,1,2] => ?
 => ? = 0 + 1
[6,5,3,4,7,8,1,2] => ?
 => ? = 2 + 1
[8,6,5,7,4,2,1,3] => ?
 => ? = 2 + 1
[8,6,7,4,5,2,1,3] => ?
 => ? = 1 + 1
[7,6,8,4,5,2,1,3] => ?
 => ? = 1 + 1
[8,7,4,5,6,2,1,3] => ?
 => ? = 2 + 1
[8,6,5,4,7,2,1,3] => ?
 => ? = 3 + 1
[8,6,4,5,7,2,1,3] => ?
 => ? = 2 + 1
[7,5,6,4,8,2,1,3] => ?
 => ? = 1 + 1
[7,5,6,4,8,1,2,3] => ?
 => ? = 1 + 1
[7,6,4,5,8,1,2,3] => ?
 => ? = 2 + 1
[7,4,5,6,8,1,2,3] => ?
 => ? = 1 + 1
[6,4,5,7,8,1,2,3] => ?
 => ? = 1 + 1
[8,6,7,4,2,3,1,5] => ?
 => ? = 1 + 1
[8,7,4,3,5,2,1,6] => ?
 => ? = 3 + 1
[7,8,5,3,4,1,2,6] => ?
 => ? = 0 + 1
[8,7,5,2,3,1,4,6] => ?
 => ? = 3 + 1
[8,5,4,6,3,2,1,7] => ?
 => ? = 2 + 1
[8,6,4,3,5,2,1,7] => ?
 => ? = 3 + 1
[8,6,4,5,2,3,1,7] => ?
 => ? = 2 + 1
[8,6,4,3,2,5,1,7] => ?
 => ? = 4 + 1
[8,6,4,5,2,1,3,7] => ?
 => ? = 2 + 1
[7,4,5,3,6,2,1,8] => ?
 => ? = 1 + 1
[7,4,5,6,2,3,1,8] => ?
 => ? = 1 + 1
[7,5,4,2,3,6,1,8] => ?
 => ? = 3 + 1
[7,4,5,6,3,1,2,8] => ?
 => ? = 1 + 1
[7,5,6,3,4,1,2,8] => ?
 => ? = 1 + 1
[7,5,3,4,6,1,2,8] => ?
 => ? = 2 + 1
[6,4,3,5,7,1,2,8] => ?
 => ? = 2 + 1
[7,4,5,6,2,1,3,8] => ?
 => ? = 1 + 1
[7,5,6,4,1,2,3,8] => ?
 => ? = 1 + 1
[7,6,4,5,1,2,3,8] => ?
 => ? = 2 + 1
[7,5,4,6,1,2,3,8] => ?
 => ? = 2 + 1
[7,5,6,3,1,2,4,8] => ?
 => ? = 1 + 1
[7,5,3,4,1,2,6,8] => ?
 => ? = 2 + 1
[6,3,2,1,4,5,7,8] => ?
 => ? = 3 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001392
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001392: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
 => [1]
 => 0
[2,1] => [2,1] => [1,1,0,0]
 => []
 => ? = 1
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 0
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => ? = 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => ? = 2
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 2
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 3
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3
[5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3
[5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 4
[6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,3,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,4,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,4,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,5,3,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,5,4,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,2,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,2,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,4,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,4,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,5,2,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,5,4,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,2,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,2,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,3,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,3,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,5,2,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St001918
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
 => [1]
 => 0
[2,1] => [2,1] => [1,1,0,0]
 => []
 => ? = 1
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 0
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => ? = 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => ? = 2
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 2
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 3
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 1
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3
[5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1
[5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3
[5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 4
[6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,3,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,4,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,4,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,5,3,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,2,5,4,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,2,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,2,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,4,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,4,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,5,2,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,3,5,4,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,2,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,2,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,3,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,3,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
[6,1,4,5,2,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition. Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$. The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is $$ \sum_{p\in\lambda} [p]_{q^{N/p}}, $$ where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer. This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals $$ \left(1 - \frac{1}{\lambda_1}\right) N, $$ where $\lambda_1$ is the largest part of $\lambda$. The statistic is undefined for the empty partition.
Matching statistic: St000667
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
 => [1]
 => 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
 => []
 => ? = 1 + 1
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 2 = 1 + 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 1 = 0 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => ? = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => ? = 2 + 1
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 3 = 2 + 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 = 0 + 1
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 3 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 4 + 1
[6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,3,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,4,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,4,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,5,3,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,5,4,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,2,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,2,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,4,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,4,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,5,2,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,5,4,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,2,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,2,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,3,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,3,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,5,2,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000668
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
 => [1]
 => ? = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
 => []
 => ? = 1 + 1
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 2 = 1 + 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => ? = 0 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => ? = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => ? = 2 + 1
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 3 = 2 + 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 0 + 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 0 + 1
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 3 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,3,1,4,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[4,5,1,2,3] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,2,1,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,2,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 4 + 1
[5,6,1,2,3,4] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,2,4,3] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,3,2,4] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,3,4,2] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,4,2,3] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,4,3,2] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,2,1,3,4] => [5,6,2,1,4,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000770
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
 => [1]
 => ? = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
 => []
 => ? = 1 + 1
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 2 = 1 + 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => ? = 0 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => ? = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => ? = 2 + 1
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 3 = 2 + 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 0 + 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => ? = 0 + 1
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 3 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,3,1,4,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[4,5,1,2,3] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,1,3,2] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,2,1,3] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,2,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,3,1,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 4 + 1
[5,6,1,2,3,4] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,2,4,3] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,3,2,4] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,3,4,2] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,4,2,3] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,1,4,3,2] => [5,6,1,4,3,2] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
[5,6,2,1,3,4] => [5,6,2,1,4,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => ? = 0 + 1
Description
The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St001571
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
 => [1]
 => 1 = 0 + 1
[2,1] => [2,1] => [1,1,0,0]
 => []
 => ? = 1 + 1
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 2 = 1 + 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 1 = 0 + 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => ? = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => ? = 2 + 1
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1 = 0 + 1
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 1 = 0 + 1
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 3 = 2 + 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 2 = 1 + 1
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 1 = 0 + 1
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 1 + 1
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => ? = 3 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 1 = 0 + 1
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 2 = 1 + 1
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 1 + 1
[5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 2 + 1
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ? = 4 + 1
[6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,3,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,4,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,4,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,5,3,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,2,5,4,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,2,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,2,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,4,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,4,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,5,2,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,3,5,4,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,2,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,2,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,3,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,3,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
[6,1,4,5,2,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => ? = 1 + 1
Description
The Cartan determinant of the integer partition. Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$. Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000678The number of up steps after the last double rise of a Dyck path. St000383The last part of an integer composition. St000439The position of the first down step of a Dyck path. St000759The smallest missing part in an integer partition. St001050The number of terminal closers of a set partition. St000617The number of global maxima of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000363The number of minimal vertex covers of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001322The size of a minimal independent dominating set in a graph. St001316The domatic number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000234The number of global ascents of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000989The number of final rises of a permutation. St000501The size of the first part in the decomposition of a permutation. St000654The first descent 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. St000542The number of left-to-right-minima of a permutation. St000260The radius of a connected graph. St000054The first entry of the permutation. St000259The diameter of a connected graph. St000990The first ascent of a permutation. St000546The number of global descents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000991The number of right-to-left minima of a permutation. 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. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000454The largest eigenvalue of a graph if it is integral.