Your data matches 140 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001263
Mapping
St001263: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 1
[1,2] => 0
[2,1] => 0
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 0
[1,2,1] => 1
[1,3] => 0
[2,1,1] => 0
[2,2] => 1
[3,1] => 0
[4] => 1
[1,1,1,1,1] => 2
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 0
[1,3,1] => 2
[1,4] => 0
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 0
[2,3] => 0
[3,1,1] => 1
[3,2] => 0
[4,1] => 0
[5] => 2
[1,1,1,1,1,1] => 2
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 2
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 0
[1,2,1,1,1] => 1
[1,2,1,2] => 0
[1,2,2,1] => 2
[1,2,3] => 0
[1,3,1,1] => 1
[1,3,2] => 0
[1,4,1] => 2
[1,5] => 0
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 0
Description
The index of the maximal parabolic seaweed algebra associated with the composition. Let $a_1,\dots,a_m$ and $b_1,\dots,b_t$ be a pair of compositions of $n$. The meander associated to this pair is obtained as follows: * place $n$ dots on a horizontal line * subdivide the dots into $m$ blocks of sizes $a_1, a_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc above the line * subdivide the dots into $t$ blocks of sizes $b_1, b_2,\dots$ * within each block, connect the first and the last dot, the second and the next to last, and so on, with an arc below the line By [1, thm.5.1], the index of the seaweed algebra associated to the pair of compositions is $$ \operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m} = 2C+P-1, $$ where $C$ is the number of cycles (of length at least $2$) and P is the number of paths in the meander. This statistic is $\operatorname{ind}\displaystyle\frac{b_1|b_2|...|b_t}{n}$.
Matching statistic: St001414
(load all 20 compositions to match this statistic)
Mapping
Mp00094: Integer compositions —to binary word⟶ Binary words
St001414: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0
[1,1] => 11 => 0
[2] => 10 => 0
[1,1,1] => 111 => 1
[1,2] => 110 => 0
[2,1] => 101 => 1
[3] => 100 => 0
[1,1,1,1] => 1111 => 1
[1,1,2] => 1110 => 1
[1,2,1] => 1101 => 0
[1,3] => 1100 => 0
[2,1,1] => 1011 => 1
[2,2] => 1010 => 1
[3,1] => 1001 => 0
[4] => 1000 => 0
[1,1,1,1,1] => 11111 => 2
[1,1,1,2] => 11110 => 1
[1,1,2,1] => 11101 => 1
[1,1,3] => 11100 => 1
[1,2,1,1] => 11011 => 2
[1,2,2] => 11010 => 0
[1,3,1] => 11001 => 0
[1,4] => 11000 => 0
[2,1,1,1] => 10111 => 1
[2,1,2] => 10110 => 1
[2,2,1] => 10101 => 2
[2,3] => 10100 => 1
[3,1,1] => 10011 => 0
[3,2] => 10010 => 0
[4,1] => 10001 => 2
[5] => 10000 => 0
[1,1,1,1,1,1] => 111111 => 2
[1,1,1,1,2] => 111110 => 2
[1,1,1,2,1] => 111101 => 1
[1,1,1,3] => 111100 => 1
[1,1,2,1,1] => 111011 => 1
[1,1,2,2] => 111010 => 1
[1,1,3,1] => 111001 => 1
[1,1,4] => 111000 => 1
[1,2,1,1,1] => 110111 => 2
[1,2,1,2] => 110110 => 2
[1,2,2,1] => 110101 => 0
[1,2,3] => 110100 => 0
[1,3,1,1] => 110011 => 0
[1,3,2] => 110010 => 0
[1,4,1] => 110001 => 0
[1,5] => 110000 => 0
[2,1,1,1,1] => 101111 => 1
[2,1,1,2] => 101110 => 1
[2,1,2,1] => 101101 => 1
Description
Half the length of the longest odd length palindromic prefix of a binary word. More precisely, this statistic is the largest number $k$ such that the word has a palindromic prefix of length $2k+1$.
Matching statistic: St001115
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => [1] => 0
[1,1] => [1,0,1,0]
 => [1,1,0,0]
 => [1,2] => 0
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [2,1] => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,3,2] => 1
[1,2] => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 0
[2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[3] => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4,6] => 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4,6] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,3,2,5,6,4] => 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,3,5,6,2,4] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,3,2,6,4,5] => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,3,6,2,4,5] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,2,4,5,6] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,3,4,5,6,2] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,2,5,3,6] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,4,5,2,3,6] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,4,2,5,6,3] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,4,5,6,2,3] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,5,2,6,3,4] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,5,6,2,3,4] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,6,2,3,4,5] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,6] => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4,6] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,3,1,5,6,4] => 1
Description
The number of even descents of a permutation.
Matching statistic: St001413
(load all 2 compositions to match this statistic)
Mapping
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0 => 0 => 0
[1,1] => 11 => 00 => 01 => 0
[2] => 10 => 01 => 10 => 0
[1,1,1] => 111 => 000 => 011 => 0
[1,2] => 110 => 001 => 101 => 0
[2,1] => 101 => 010 => 000 => 1
[3] => 100 => 011 => 110 => 1
[1,1,1,1] => 1111 => 0000 => 0111 => 0
[1,1,2] => 1110 => 0001 => 1011 => 0
[1,2,1] => 1101 => 0010 => 0001 => 1
[1,3] => 1100 => 0011 => 1101 => 1
[2,1,1] => 1011 => 0100 => 0100 => 0
[2,2] => 1010 => 0101 => 1000 => 0
[3,1] => 1001 => 0110 => 0010 => 1
[4] => 1000 => 0111 => 1110 => 1
[1,1,1,1,1] => 11111 => 00000 => 01111 => 0
[1,1,1,2] => 11110 => 00001 => 10111 => 0
[1,1,2,1] => 11101 => 00010 => 00011 => 1
[1,1,3] => 11100 => 00011 => 11011 => 1
[1,2,1,1] => 11011 => 00100 => 01001 => 0
[1,2,2] => 11010 => 00101 => 10001 => 0
[1,3,1] => 11001 => 00110 => 00101 => 1
[1,4] => 11000 => 00111 => 11101 => 1
[2,1,1,1] => 10111 => 01000 => 01100 => 2
[2,1,2] => 10110 => 01001 => 10100 => 0
[2,2,1] => 10101 => 01010 => 00000 => 2
[2,3] => 10100 => 01011 => 11000 => 1
[3,1,1] => 10011 => 01100 => 01010 => 0
[3,2] => 10010 => 01101 => 10010 => 2
[4,1] => 10001 => 01110 => 00110 => 1
[5] => 10000 => 01111 => 11110 => 2
[1,1,1,1,1,1] => 111111 => 000000 => 011111 => 0
[1,1,1,1,2] => 111110 => 000001 => 101111 => 0
[1,1,1,2,1] => 111101 => 000010 => 000111 => 1
[1,1,1,3] => 111100 => 000011 => 110111 => 1
[1,1,2,1,1] => 111011 => 000100 => 010011 => 0
[1,1,2,2] => 111010 => 000101 => 100011 => 0
[1,1,3,1] => 111001 => 000110 => 001011 => 1
[1,1,4] => 111000 => 000111 => 111011 => 1
[1,2,1,1,1] => 110111 => 001000 => 011001 => 2
[1,2,1,2] => 110110 => 001001 => 101001 => 0
[1,2,2,1] => 110101 => 001010 => 000001 => 2
[1,2,3] => 110100 => 001011 => 110001 => 1
[1,3,1,1] => 110011 => 001100 => 010101 => 0
[1,3,2] => 110010 => 001101 => 100101 => 2
[1,4,1] => 110001 => 001110 => 001101 => 1
[1,5] => 110000 => 001111 => 111101 => 2
[2,1,1,1,1] => 101111 => 010000 => 011100 => 0
[2,1,1,2] => 101110 => 010001 => 101100 => 0
[2,1,2,1] => 101101 => 010010 => 000100 => 1
Description
Half the length of the longest even length palindromic prefix of a binary word.
Matching statistic: St001355
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St001355: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => => ? = 0
[1,1] => [1,0,1,0]
 => [1,2] => 0 => 0
[2] => [1,1,0,0]
 => [2,1] => 1 => 0
[1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 00 => 0
[1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 01 => 1
[2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 10 => 1
[3] => [1,1,1,0,0,0]
 => [3,2,1] => 11 => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 000 => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 001 => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 010 => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 011 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 100 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 101 => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 110 => 0
[4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 111 => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0000 => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0001 => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0010 => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0011 => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0100 => 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0101 => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0110 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 0111 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1000 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1001 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1010 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1011 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1100 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1101 => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1110 => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1111 => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 00000 => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => 00001 => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => 00010 => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,5,4] => 00011 => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => 00100 => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => 00101 => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => 00110 => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,5,4,3] => 00111 => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => 01000 => 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => 01001 => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => 01010 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,6,5,4] => 01011 => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => 01100 => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,4,3,2,6,5] => 01101 => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => 01110 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 01111 => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 10000 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => 10001 => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 10010 => 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3,6,5,4] => 10011 => 2
Description
Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
Matching statistic: St000137
(load all 4 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
 => []
 => ? = 0
[1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,0}
[2] => [[2],[]]
 => []
 => ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
 => [1]
 => 1
[3] => [[3],[]]
 => []
 => ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,1,1}
[1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,1,1}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
[1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,1,1}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => 0
[4] => [[4],[]]
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
[1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
[1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[2,3] => [[4,2],[1]]
 => [1]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
[3,2] => [[4,3],[2]]
 => [2]
 => 0
[4,1] => [[4,4],[3]]
 => [3]
 => 1
[5] => [[5],[]]
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0
[1,1,4] => [[4,1,1],[]]
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 2
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => 0
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => 1
[1,5] => [[5,1],[]]
 => []
 => ? ∊ {0,0,2,2,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 0
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 0
[2,4] => [[5,2],[1]]
 => [1]
 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 2
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => 0
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 2
[4,2] => [[5,4],[3]]
 => [3]
 => 1
[5,1] => [[5,5],[4]]
 => [4]
 => 0
[6] => [[6],[]]
 => []
 => ? ∊ {0,0,2,2,2,2}
Description
The Grundy value of an integer partition. Consider the two-player game on an integer partition. In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition. The first player that cannot move lose. This happens exactly when the empty partition is reached. The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1]. This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St000329
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
 => []
 => []
 => ? = 0
[1,1] => [[1,1],[]]
 => []
 => []
 => ? ∊ {0,0}
[2] => [[2],[]]
 => []
 => []
 => ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {0,1,1}
[1,2] => [[2,1],[]]
 => []
 => []
 => ? ∊ {0,1,1}
[2,1] => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[3] => [[3],[]]
 => []
 => []
 => ? ∊ {0,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,1,1}
[1,1,2] => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,1,1}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,3] => [[3,1],[]]
 => []
 => []
 => ? ∊ {0,0,1,1}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4] => [[4],[]]
 => []
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,3] => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,4] => [[4,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2] => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[5] => [[5],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,4] => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,5] => [[5,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[4,2] => [[5,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[5,1] => [[5,5],[4]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[6] => [[6],[]]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0}
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000658
(load all 2 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
 => []
 => []
 => ? = 0
[1,1] => [[1,1],[]]
 => []
 => []
 => ? ∊ {0,0}
[2] => [[2],[]]
 => []
 => []
 => ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {0,1,1}
[1,2] => [[2,1],[]]
 => []
 => []
 => ? ∊ {0,1,1}
[2,1] => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[3] => [[3],[]]
 => []
 => []
 => ? ∊ {0,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,1,1}
[1,1,2] => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,1,1}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,3] => [[3,1],[]]
 => []
 => []
 => ? ∊ {0,0,1,1}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4] => [[4],[]]
 => []
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,3] => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {1,1,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,4] => [[4,1],[]]
 => []
 => []
 => ? ∊ {1,1,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2] => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[5] => [[5],[]]
 => []
 => []
 => ? ∊ {1,1,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,2,2,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,2,2,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,2,2,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,4] => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {2,2,2,2,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,5] => [[5,1],[]]
 => []
 => []
 => ? ∊ {2,2,2,2,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[4,2] => [[5,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[5,1] => [[5,5],[4]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[6] => [[6],[]]
 => []
 => []
 => ? ∊ {2,2,2,2,2,2}
Description
The number of rises of length 2 of a Dyck path. This is also the number of $(1,1)$ steps of the associated Łukasiewicz path, see [1]. A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St000932
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
 => []
 => []
 => ? = 0
[1,1] => [[1,1],[]]
 => []
 => []
 => ? ∊ {0,0}
[2] => [[2],[]]
 => []
 => []
 => ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
 => []
 => []
 => ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3] => [[3],[]]
 => []
 => []
 => ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,1}
[1,1,2] => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,1}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3] => [[3,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,1}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4] => [[4],[]]
 => []
 => []
 => ? ∊ {0,0,0,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,3] => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,4] => [[4,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,1] => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[5] => [[5],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,4] => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,5] => [[5,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[4,2] => [[5,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[5,1] => [[5,5],[4]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[6] => [[6],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
Description
The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001067
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
 => []
 => []
 => ? = 0
[1,1] => [[1,1],[]]
 => []
 => []
 => ? ∊ {0,0}
[2] => [[2],[]]
 => []
 => []
 => ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
 => []
 => []
 => ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3] => [[3],[]]
 => []
 => []
 => ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,1}
[1,1,2] => [[2,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,1}
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3] => [[3,1],[]]
 => []
 => []
 => ? ∊ {0,0,0,1}
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1] => [[3,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4] => [[4],[]]
 => []
 => []
 => ? ∊ {0,0,0,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,3] => [[3,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,4] => [[4,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[3,2] => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,1] => [[4,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[5] => [[5],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,1,4] => [[4,1,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,5] => [[5,1],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 2
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0,1,0]
 => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 0
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 0
[4,2] => [[5,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0
[5,1] => [[5,5],[4]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[6] => [[6],[]]
 => []
 => []
 => ? ∊ {0,0,2,2,2,2}
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
The following 130 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001877Number of indecomposable injective modules with projective dimension 2. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000260The radius of a connected graph. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000938The number of zeros of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000741The Colin de Verdière graph invariant. St000941The number of characters of the symmetric group whose value on the partition is even. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001525The number of symmetric hooks on the diagonal of a partition. St001587Half of the largest even part of an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching. St001651The Frankl number of a lattice. St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000770The major index of an integer partition when read from bottom to top. St000781The number of proper colouring schemes of a Ferrers diagram. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000944The 3-degree of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001175The size of a partition minus the hook length of the base cell. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001556The number of inversions of the third entry of a permutation. St001586The number of odd parts smaller than the largest even part in an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001960The number of descents of a permutation minus one if its first entry is not one. St001961The sum of the greatest common divisors of all pairs of parts. St000630The length of the shortest palindromic decomposition of a binary word. St001569The maximal modular displacement of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001568The smallest positive integer that does not appear twice in the partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000089The absolute variation of a composition. St000365The number of double ascents of a permutation. St000650The number of 3-rises of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001822The number of alignments of a signed permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001768The number of reduced words of a signed permutation. St001964The interval resolution global dimension of a poset. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph.