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Name First elements ... The number of free polyominoes with n cells. A000105: Catalan numbers C n: ... Numbers of the form pq where p and q are distinct primes ...
The final result is that the last cell contains all the longest subsequences common to (AGCAT) and (GAC); these are (AC), (GC), and (GA). The table also shows the longest common subsequences for every possible pair of prefixes. For example, for (AGC) and (GA), the longest common subsequence are (A) and (G).
For example, the sequence ,, is a subsequence of ,,,,, obtained after removal of elements ,, and . The relation of one sequence being the subsequence of another is a partial order . Subsequences can contain consecutive elements which were not consecutive in the original sequence.
There is no similar relationship between shortest common supersequences and longest common subsequences of three or more input sequences. (In particular, LCS and SCS are not dual problems .) However, both problems can be solved in O ( n k ) {\displaystyle O(n^{k})} time using dynamic programming, where k {\displaystyle k} is the number of ...
In general there are many sequences for a particular n and k but in this example it is unique, up to cycling. In combinatorial mathematics, a de Bruijn sequence of order n on a size-k alphabet A is a cyclic sequence in which every possible length-n string on A occurs exactly once as a substring (i.e., as a contiguous subsequence).
This subsequence has length six; the input sequence has no seven-member increasing subsequences. The longest increasing subsequence in this example is not the only solution: for instance, 0, 4, 6, 9, 11, 15 0, 2, 6, 9, 13, 15 0, 4, 6, 9, 13, 15. are other increasing subsequences of equal length in the same input sequence.
For example, for the array of values [−2, 1, −3, 4, −1, 2, 1, −5, 4], the contiguous subarray with the largest sum is [4, −1, 2, 1], with sum 6. Some properties of this problem are: If the array contains all non-negative numbers, then the problem is trivial; a maximum subarray is the entire array.
For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1. The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1.