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Problem · Dynamic Programming

Number of Balanced Strings

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Problem statement

Count the balanced strings of length n over the lowercase English alphabet.

A string is balanced when the absolute difference between the alphabet positions of every pair of adjacent characters is at most d.

Return the count modulo 1,000,000,007.

Function

numberOfBalancedStrings(n: int, d: int) → int

Examples

Example 1

n = 3d = 3return = 1134

Dynamic counting over the final letter gives 1,134 balanced lowercase strings of length 3 when adjacent letters may differ by at most 3.

Example 2

n = 2d = 2return = 124

There are 124 ordered pairs of lowercase letters whose alphabet-position difference is at most 2.

Constraints

1 <= n <= 10^6
0 <= d <= 25