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1

Let $\Sigma$ be your alphabet, and $w \in \Sigma^*$ be your word. Let $w_i$ be the word consisting of the first $i$ characters of $w$ (for $i \le 0$, $w_i$ is the empty word $\varepsilon$). For $c \in \Sigma$, let $S[i,k,c]$ be the number of subsequences of $w_i$ that end with character $c$ and have length $k$. Moreover, for a word $w'$, let $\ell_c(w')$ be ...

2

Let's solve your example question first. Say, we want to know the number of combinations of size 2 possible from $A, B, C$. This would be equal to for each element in $A$, pair it with each element in $B$. This gives us $|N_A||N_B|$ combinations of size 2. for each element in $A$, pair it with each element in $C$. This gives us $|N_A||N_C|$ combinations of ...

2

If I understand the question, your problem is the following: Given a regular language $L$ over alphabet $\Sigma$ and a positive integer $n$, compute the probability that a word chosen uniformly at random from $\Sigma^n$ will be in $L$. That's equivalent to computing $|L \cap \Sigma^n|$, i.e., the cardinality of the language $L \cap \Sigma^n$. Note that $... 1 Every$m$-ary tree is a node together with up to$m$children, which are also$m$-ary trees. Let$T(x,y)$be the generating function in which the coefficient of$x^ny^k$is the number of$m$-ary trees with$n$nodes and$k$full nodes. Then $$T(x,y) = x(1 + T(x,y) + \cdots + T(x,y)^{m-1} + yT(x,y)^m).$$ The generating function of the total number of$m$-... 1 Your problem is related to the balanced tournament design problem. Suppose that you have$2n$tennis players, there are$n$tennis courts, and you need to play a round robin tournament in$2n-1\$ rounds. The courts aren't the same, to to balance the effect of the different courts, you want to ensure that no player competes more than twice on any one court. ...

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As pointed out by @Pseudonym, this problem is identical to finding a round robin tournament schedule. A straightforward implementation would be the following: def get_sets(a: list): # Simple implementation of # https://nrich.maths.org/1443 center, *rest = a for _ in range(len(rest)): rest = rest[1:] + rest[:1] # put 1st element to ...

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