Consider a generalized deck of $n$ cards. That is, $n$ objects (generalized cards) characterized by a pair of discrete indices: $(i,j)$, where $i$ is the rank and $i=1,\dots,R$ and $j$ is the suit and $j=1,\dots,S$.
For a regular deck we have $R=13$ (Ace to King) and $S=4$ (Spades to Hearts). In addition we consider a $T$-"tuple" of cards with $T \leq S$. A $T$-tuple is $T$ (or more) cards of the same rank (same i index) in a row within the shuffled deck.
Question: How many arrangements of these cards n have $T$-tuples in them?
intuitive: My question originated from the special case of a regular deck and $T=2$ i.e. pairs. How many arrangements of the $52$ cards have no pairs in them? (and from there one can easily calculate the probability of finding no pairs in a shuffled deck) for example a shuffled deck that goes like 8 3 7 4 6 7 5 4 K Q K K ...etc hits a pair in the last 2 cards