Derangement count: sort four-card orders by cycle type

Derangement count asks how many orders leave every label away from its matching position. With four distinct labels and four fixed named positions, there are nine such orders. We can count them by separating two possible cycle types: three orders consist of two pair swaps, and six consist of one four-cycle.

This is an invented paper exercise. The labels A, B, C and D identify abstract cards, not people, gift recipients, product variants or delivered quantities. Each label is used exactly once. The position names also remain fixed throughout the count, so physically rotating a written row does not make it the same assignment.

Declare the positions and the row convention

Write four columns headed position A, position B, position C and position D. The original row has A, B, C and D in those respective columns. An order such as BADC means that position A contains label B, position B contains A, position C contains D and position D contains C.

Call the lookup f: f(position name) is the label appearing at that named position. For BADC, f(A) = B and f(B) = A, while f(C) = D and f(D) = C. This convention reads a row. An arrow in the cycle notation below records that lookup; it is not an instruction to move a particular physical card.

An eligible order must satisfy two requirements. First, the four labels A, B, C and D each appear once. Second, no column contains its own name as its label. Thus f(A) must differ from A, and the same check applies independently at B, C and D. A row merely differing from ABCD is not enough.

Follow one lookup cycle

In BADC, begin at the name A and follow the lookup. A leads to B, and B leads back to A. Write this as the two-cycle (A B). It contains two names and records both lookups, including the return from the last written name to the first.

The remaining names form (C D), since C leads to D and D returns to C. The complete notation is therefore (A B)(C D). The parentheses identify two separate cycles. They do not say that A and B occupy neighboring columns or that the row can be obtained within a particular number of adjacent-card moves.

Every order using each label once has a cycle description. Starting at any name and following f must eventually revisit a name because there are only four. The first revisit closes the cycle at its starting name: entering an earlier nonstarting name would give that name two different predecessors, contrary to each label occurring only once as a lookup value.

After recording that cycle, start again from an unused name. It cannot enter an already recorded cycle without giving one of its names an additional predecessor. Continue until all four names are recorded. This produces disjoint cycles whose lengths add to four, with every lookup represented.

Restrict the possible cycle lengths

A one-cycle would send a name back to itself. For example, (D) means f(D) = D, putting label D in position D. That is exactly a matching position, so an eligible order cannot contain a one-cycle. Every cycle must contain at least two names.

Four names can consequently split only as two plus two or as four. A three-cycle would leave one name by itself, which is forbidden. A two-cycle with two names left separately would also contain forbidden one-cycles. These length possibilities establish a complete classification before we begin adding counts.

The two allowed types are disjoint. In a two-pair order, following the lookup from A returns to A after two steps. In a four-cycle order, the same walk first returns after four. One fixed lookup cannot have both of those first-return lengths, so an order cannot belong to both groups.

Count the three pair-swap orders

For the two-plus-two type, choose A's partner. It can be B, C or D, giving three choices. Once that pair is fixed, the two remaining names must form the other pair. There is no further partner choice because only those two names remain.

Pairing A with B gives (A B)(C D), whose row is BADC. Pairing A with C gives (A C)(B D), whose row is CDAB. Pairing A with D gives (A D)(B C), whose row is DCBA. Check each row under the fixed A-B-C-D column headings: every label occurs once and no label matches its column.

Each pair-swap order has exactly one partner for A, so these three choices cannot produce the same row. They also cover every two-plus-two order, since A must belong to one of its two pairs. This supplies both the count of three and a reason the list is complete.

Changing how the pairs are written does not create another row. For example, (B A)(D C) still gives BADC, as does writing the C-D cycle before the A-B cycle. Count the lookup assignment once. The partner-of-A method avoids treating the order of the written parentheses as another choice.

Count the six four-cycles

For a single four-cycle, write A first. Arrange B, C and D after it in any order. There are three choices for the second written name, two for the third and one for the fourth. Multiplying gives 3 times 2 times 1, or six anchored cycle descriptions.

The cycle (A B C D) means A leads to B, B to C, C to D and D back to A. Reading those values in fixed column order gives BCDA. The cycle (A B D C) instead gives BDAC. Both use every name in one four-cycle, and neither places a label in its matching column.

The two descriptions beginning A-C are (A C B D), giving CDBA, and (A C D B), giving CADB. The two beginning A-D are (A D B C), giving DCAB, and (A D C B), giving DABC. These six rows are all different.

Conversely, a four-cycle lookup determines one of these descriptions once A is chosen as its written start. Follow A's value, then the next value, then the next, before returning to A. The resulting order of the other three names is fixed by the lookup. Thus the six written descriptions neither miss nor duplicate a four-cycle assignment.

Keep notation changes separate from row changes

Writing (B C D A) instead of (A B C D) preserves every lookup and still describes BCDA. Choosing A as the written start is a convention that gives one description per four-cycle. It does not allow us to rotate labels among the four named columns.

For a concrete distinction, rotate the valid printed row BADC one place left. It becomes ADCB. Position A now contains A, and position C contains C, so the rotated row is ineligible. The columns have stayed fixed while their contents changed. Treating these two printed rows as interchangeable would change the problem.

Reversing a cycle's direction also changes its lookup. (A D C B) gives DABC, whereas (A B C D) gives BCDA. Both happen to be valid four-cycle orders and are included separately among the six. A changed starting point and a reversed direction therefore need different treatment in the count.

Combine the complete lists

The two-pair rows are BADC, CDAB and DCBA. The four-cycle rows are BCDA, BDAC, CDBA, CADB, DCAB and DABC. Adding the disjoint counts gives 3 + 6 = 9 eligible orders. The cycle-length classification establishes why there is no third allowed type left to count.

As a separate size check, four distinct labels have 4 times 3 times 2 times 1, or 24 total orders before restrictions. Nine is a count within that larger set. No random assignment rule or equally likely shuffle has been stipulated here, so the result is being reported as an order count, not a probability of a real event.

Check the input conditions on rejected rows

BCAD contains the three-cycle (A B C) and the one-cycle (D). It fails because D remains in position D. BACD contains the pair (A B) but leaves C and D fixed. A successful swap in one part of a row does not establish that all four position checks pass.

BAAC illustrates another failure. None of its four labels matches its column name, but A occurs twice and D is absent. It is not an order of the required four distinct labels used once each. The cycle classification assumed a bijection from the start, so the no-match check cannot replace that input requirement.

Keep the nine valid rows with the original column headings and both eligibility rules. Someone checking the worksheet can recover each cycle, inspect every column and reproduce the three-plus-six count. If the labels cease to be distinct or the position identity rule changes, start a separately defined problem rather than carrying nine into it.

Read native tag accounts separately

The native QCP merchandise-tag account shows a red-and-white finished tag with a hole and attached black loop. Special Sale, WAS and NOW are pictured artwork, not evidence of a current price or discount. Maker, model, material, dimensions, delivered quantity and attachment performance remain unverified.

The native No. 9 merchandise-tag account shows a finished white scalloped tag with attached string above a box. Its description has malformed dimensions and an incomplete box count; the pictured QTY 1000 label does not verify delivered quantity. Cardstock is its catalog category. Maker, material and performance remain unverified.

Neither account supplies the four abstract labels or the counting proof. The No. 9 catalog designation does not establish the mathematical total of nine orders. The exercise requires no purchase and includes no instruction to attach, print on or otherwise use either tag.

Distinguish other order questions

The inverse position map guide constructs a return map for one supplied permutation. Here the task is to count all eligible four-label assignments. The declared lookup direction makes cycle notation readable, but no inverse-restoration procedure supplies the nine-order count.

The holiday ring rotation guide identifies unmarked circular color patterns after turns. This exercise keeps named positions fixed. Only the written start of a function cycle can change without changing its lookups; physically rotating the card row may change its eligibility.

Keep any checkout enquiry outside the count

If you independently consider a tag order on republeq.com and want to ask about USDT, begin with the exact offer and intended quantity. The labels and counts in this worksheet are invented mathematical inputs, not product dimensions, prices, cryptocurrency amounts or payment instructions. Correct arithmetic establishes neither a currently offered payment method nor the status of an actual order. Read the choices and conditions presented for your intended transaction, and ask the responsible support contact to clarify essential missing information before deciding whether to proceed. If a cryptocurrency option is offered, establish the specified asset and network, requested amount, any stated validity period and the reference associating a payment with the order. Do not infer a transfer destination from a product photograph, a drawing or an earlier transaction. Resolve product and supplied-content questions separately from payment instructions. Ask about applicable delivery, cancellation, return and refund conditions, including the denomination of any approved refund and relevant deductions. Keep each answer attached to the offer it concerns. Information about one tag account does not automatically describe the other, and a changed quantity may require reviewing the resulting order information again. This article promises no present USDT availability, fee level, exchange result, confirmation time, refund amount or delivery outcome. You can inspect every written order without purchasing anything. Preserve any eventual order acknowledgement and transaction record separately from this worksheet. Those records concern an actual transaction under its documented conditions; the card count follows expressly declared rules. A correct nine-order count cannot resolve an unanswered checkout question.