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Random Monotone Grid Route

Replay seed or one advancing injected source, uniform over the explicitly stated finite support, not cryptographically secure. Sample uniformly over monotone route move sequences by choosing down-step positions; no obstacle pathfinding, speed or geometric shortest-path guarantee beyond grid moves.

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Description

Sample uniformly over monotone route move sequences by choosing down-step positions; no obstacle pathfinding, speed or geometric shortest-path guarantee beyond grid moves.

A random construction is defined by its sampling support: positions, permutations, complete tilings, proposal sequences or another explicit set. Uniform choices at one stage do not necessarily make final visible outcomes uniform. A replay seed gives a deterministic sequence; equal values can hide different sampled positions. These tools are for reproducible synthetic examples rather than cryptographic entropy.

Sample uniformly over monotone route move sequences by choosing down-step positions; no obstacle pathfinding, speed or geometric shortest-path guarantee beyond grid moves.

When to use this tool

Use this tool to sample uniformly over monotone route move sequences by choosing down-step positions; no obstacle pathfinding, speed or geometric shortest-path guarantee beyond grid moves.

How to use the tool and how it works

  1. Enter right, down, seed using the definitions below.
  2. Run the operation. Resolve input-format errors before interpreting an output.
  3. Read result.seed, result.moves, result.points, result.support, using the stated boundary, ordering, sign and absence conventions.

Require positive total right+down steps no greater than 127. Choose the down-step positions uniformly without replacement, fill other slots with right and accumulate grid points from [0,0]. Return C(right+down,down) as exact support text.1

Input fieldMeaning and unitsAccepted input
rightInteger number of right moves, 0–127; right+down must be 1–127.Finite safe integer satisfying the stated bounds.
downInteger number of down moves, 0–127; combined move count must be 1–127.Finite safe integer satisfying the stated bounds.
seedNonblank replay seed, at most 256 UTF-16 code units; an injected source overrides seeded draws.Text, at most 8,192 UTF-16 code units; narrower grammar/range rules also apply.
Output fieldInterpretation
resultObject containing the fields below.
result.seedOriginal supplied seed string; injected-source replay additionally requires that source’s state.
result.movesGenerated right/down move labels in route order.
result.pointsRoute vertices including the initial [0,0] and final endpoint.
result.supportExact finite support count as decimal text.
json
{
  "input": {
    "right": 2,
    "down": 1,
    "seed": "batch56"
  },
  "output": {
    "result": {
      "seed": "batch56",
      "moves": [
        "right",
        "down",
        "right"
      ],
      "points": [
        [
          0,
          0
        ],
        [
          1,
          0
        ],
        [
          1,
          1
        ],
        [
          2,
          1
        ]
      ],
      "support": "3"
    }
  }
}

Limitations and assumptions

  • All routes are monotone in positive right/down coordinates; there is no obstacle, wrap or shortest-path search beyond this fixed move bank.
  • A nonblank seed of at most 256 UTF-16 code units is required. With ordinary use the seed initializes one deterministic advancing source; an application-injected random source takes precedence and the seed alone then cannot replay the result. Uniformity statements assume independent uniform source draws in [0,1). Source calls are bounded and degenerate injected sources can fail a rejection budget. This is not cryptographically secure.
  • Every declared outer field is required and unknown outer fields are rejected. Strings and JSON text have size limits, lists are bounded to 128 entries, and JSON nesting/node counts are bounded. Tighter operation-specific bounds stated above take precedence.
  • Ordinary numeric outputs must be finite and within safe-integer magnitude. Exact large lengths/counts use decimal strings. Original record indexes and code-unit positions are zero-based; span ends are exclusive unless stated otherwise.

Alternative or Complementary methods

Choose a generator whose support matches the intended experiment and retain source-position provenance. Use a separate statistically justified process model or cryptographic source where those properties matter.

References

  1. Tool behavior and example: Random Monotone Grid Route — tool.io

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