2. Notation

This section is mostly meant to serve as a reference for the names and symbols introduced in what follows. However, a first brief look can serve as a good warm-up.

Denote by $c_i$ the $i$-th card on the deck. The deck is thus given by the sequence $\mathcal{C}=\{c_i{\}}_{i=1}^{98}$, of which there exist 98! distinct ones. As is discussed in this tangent though, for the purpose of this problem, a number of them are “isomorphic”, or “pretty much the same”, by mirror symmetry. Naturally, $2\le c_i\le 99$ for all $i$.
Label by $\mathcal{H}$ the hand of the player. In the case of multiple players, superscript each player by their order, i.e. ${\mathcal{H}}^i$ denotes the hand of the $i$-th player. As this is not constant, and since this is a turn-based game, the hand will vary according to each turn. We can thus illustrate this by subscripting by $j$ the hand after the $j$-th round, and call ${\mathcal{H}}_0^i$ the hand of the $i$-th player after the cards have been dealt. Note that the subscript $j$ varies according to the amount of players. Its range will be specified for the case studied. It is easy to see that for the 1-player game, we can take $j\le 46$, and for the 2-player game $j\le 28$. Note also that while in the hand, the cards behave as a set so their ordering doesn’t matter, as opposed to their position in the deck configuration $\mathcal{C}$.

Label by $U,U^{\prime }$ the two piles going up and by $D,D^{\prime }$ the two piles going down. As the order of them doesn’t have an effect in the game we don’t care about serialising them.

We refer to the set of rules of the game by $\mathcal{R}$.

In summary:

$c_i$ : the $i$-th card on the deck

$\mathcal{C}$: a configuration of the deck

${\mathcal{H}}_i^j$: the hand of the $i$-th player after the $j$-th turn. For the 1-player game, the player subscript is omitted.

$U,U^{\prime }$: the ascending piles

$D,D^{\prime }$: the descending piles

$\mathcal{R}$: the set of rules of the game

Notation from A subsequence formulation §

Summary of Notation from A subsequence formulation

$\sigma (s)$: the minimum number of subsequences that partition the sequence $s$

$L(N)$: the maximum number attained by $\sigma (s)$ for all $s$ that are permutations of the numbers $1$ to $N$

${\mathcal{R}}^{\prime }$ the alternative ruleset where no cards are drawn to begin with, and a player must play the top card of the deck as drawn.