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Labeled Unlabeled Orbit Quotient Tool

Divide an exact labeled-object count by a supplied common orbit size, requiring exact divisibility. Applicability of the common-size assumption is the caller's responsibility.

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Description

Parse exact decimal counts, require nonnegative labeled count, positive orbit size, and exact divisibility.

A counting formula requires a well-defined set of objects and an equivalence rule. Dividing a labeled count by an orbit size is justified only when the stated uniform-orbit assumption actually holds.

Parse exact decimal counts, require nonnegative labeled count, positive orbit size, and exact divisibility.

When to use this tool

Use Labeled Unlabeled Orbit Quotient when you need an exact counting component under supplied combinatorial assumptions.

How to use the tool and how it works

  1. Enter labeledCountDecimal, orbitSizeDecimal. Use the units and conventions below.
  2. Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
  3. Read unlabeledCountDecimal. Use the output convention below.

Parse exact decimal counts, require nonnegative labeled count, positive orbit size, and exact divisibility. Return the quotient as unlabeledCountDecimal.1

Input fieldMeaning and unitsAccepted input
labeledCountDecimalExact nonnegative integer count under the caller’s combinatorial model.Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above.
orbitSizeDecimalPositive common orbit size asserted by the caller; uniformity is not checked.Required. Text; maximum UTF-16 length 8192. Integer syntax: optional + or - and 1-128 decimal digits; leading zeros accepted. Apply the sign restriction above.
Output fieldInterpretationReturned shape
unlabeledCountDecimalExact integer quotient under the caller-supplied uniform-orbit assumption.Required. Text; maximum UTF-16 length 8192.
json
{
  "input": {
    "labeledCountDecimal": "120",
    "orbitSizeDecimal": "6"
  },
  "output": {
    "unlabeledCountDecimal": "20"
  }
}

Limitations and assumptions

  • Division models uniform orbit size supplied by the caller; no group action, stabilizer, equivalence relation, or equal-orbit assumption is verified.
  • Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.

Alternative or Complementary methods

Verify the action, stabilizers, and full counting argument before interpreting the quotient as a count of distinct objects.

References

  1. Calculation and input contract: combinatorics/labeled-unlabeled-orbit-quotient — tool.io

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