P13445 [GCJ 2009 #3] Alphabetomials

As we all know, there is a big difference between polynomials of degree $4$ and those of degree $5$. The question of the non-existence of a closed formula for the roots of general degree $5$ polynomials produced the famous Galois theory, which, as far as the author sees, bears no relation to our problem here. We consider only the multi-variable polynomials of degree up to $4$, over $26$ variables, represented by the set of $26$ lowercase English letters. Here is one such polynomial: $\text{aber} + \text{aab} + \text{c}$ Given a string $s$, we evaluate the polynomial on it. The evaluation gives $p(S)$ as follows: Each variable is substituted with the number of appearances of that letter in $S$. For example, take the polynomial above, and let $S = \text{"abracadabra edgar"}$. There are six a's, two b's, one c, one e, and three r's. So $p(S) = 6 \times 2 \times 1 \times 3 + 6 \times 6 \times 2 + 1 = 109.$ Given a dictionary of distinct words that consist of only lower case letters, we call a string $S$ a $d$-phrase if $S = "S_1 \ S_2 \ S_3 \ \ldots \ S_d",$ where $S_i$ is any word in the dictionary, for $1 \leqslant i \leqslant d$. i.e., $S$ is in the form of $d$ dictionary words separated with spaces. Given a number $K \leqslant 10$, your task is, for each $1 \leqslant d \leqslant K$, to compute the sum of $p(S)$ over all the $d$-phrases. Since the answers might be big, you are asked to compute the remainder when the answer is divided by $10009$.

The first line contains the number of cases $T$. $T$ test cases follow. The format of each test case is: A line containing an expression $p$ for the multi-variable polynomial, as described below in this section, then a space, then follows an integer $K$. A line with an integer $n$, the number of words in the dictionary. Then $n$ lines, each with a word, consists of only lower case letters. No word will be repeated in the same test case. We always write a polynomial in the form of a sum of terms; each term is a product of variables. We write $a^{t}$ simply as $t$ a's concatenated together. For example, $a^{2} b$ is written as $aab$. Variables in each term are always lexicographically non-decreasing.

For each test case, output a single line in the form Case #X: $sum_{1}$ $sum _{2}$ $\ldots$ $sum _{k}$ where $X$ is the case number starting from 1, and sum $_{i}$ is the sum of $p(S)$, where $S$ ranges over all $i$-phrases, modulo 10009.

**Limits** - $1 \leqslant T \leqslant 100$. - The string $p$ consists of one or more terms joined by '+'. It will not start nor end with a '+'. There will be at most 5 terms for each $p$. Each term consists at least 1 and at most 4 lower case letters, sorted in non-decreasing order. No two terms in the same polynomial will be the same. - Each word is non-empty, consists only of lower case English letters, and will not be longer than 50 characters. No word will be repeated in the same dictionary. **Small dataset** - Time limit: ~~30~~ 3 seconds. - $1 \leqslant {n} \leqslant 20$ - $1 \leqslant {K} \leqslant 5$ **Large dataset** - Time limit: ~~60~~ 6 seconds. - $1 \leqslant {n} \leqslant 100$ - $1 \leqslant {K} \leqslant 10$