Features

• Basic permutation operations: cyclic left/right shifts, prefix reversal

• Cycle analysis: find cycles in Cayley graphs with detailed information about visited states

• Sequence optimization: search for operation sequences that produce maximum cycle length

• Fast processing: lightweight version for batch testing of combinations

Package Functions

shift_left(state) — performs a cyclic left shift, moving the first element of the vector to the end.

shift_right(state) — performs a cyclic right shift, moving the last element of the vector to the beginning.

reverse_prefix(state, k) — reverses the order of the first k elements of the state vector (the “turnstile” operation in TopSpin).

apply_operations(state, operations, k) — sequentially applies a set of operations to the state. Operations are specified by symbols: “L”/“1” (left), “R”/“2” (right), “X”/“3” (reverse).

get_reachable_states(start_state, allowed_positions, k) — finds the complete cycle in the Cayley graph for a given operation sequence. Returns all visited states, a data frame with the trajectory, and cycle statistics.

get_reachable_states_light(start_state, allowed_positions, k) — lightweight version of the cycle search function. Returns only cycle length and the number of unique states without storing all intermediate states. Used for fast testing of many sequences.

find_best_random_combinations(moves, combo_length, n_samples, n_top, start_state, k) — generates random operation sequences of a given length and finds those that produce maximum cycle length. Returns the top-N best combinations with their characteristics.

Applications

The package can be useful for:

• Investigating mathematical properties of permutation puzzles

• Analyzing Cayley graphs and their structure

• Finding optimal algorithms for solving combinatorial problems

• Educational purposes in group theory and combinatorics

Dependencies

The package uses:

• digest — for efficient state hashing during cycle search