The lists below are presented in TeX math mode format with the rows of each matrix written as permutations. Each list item is numbered as a birack, biquandle, rack or quandle the prefix BR, BQ, R or Q respectively together with the number of crossings as a superscript and an index as a subscript.

The up and down actions contained in the lists on this page correspond to the maps s^{a} and s_{a}. The lists are defined up to isomorphism: permutation symmetries (conjugation), inverse (interchange of classical crossing sign), interchange of the up and down action (orientation reversal), and combinations thereof, as described in reference [1] on
the finite biquandles page.

For quandle related list items the corresponding related quandles are also shown. For example the biquandle BQ^5_{1} is quandle related, having it's up acton related to Q^5_{1} and down action related to the quandle Q^5_{1}. This is shown in the following lists as BQ^5_{1}(Q^5_{21},Q^5_{1}). Another example is birack BR^6_{7}, whose up action is not quandle related but whose down action is related to Q^6_{2}, this is shown as BR^6_{7}(-,Q^6_{2}).

Each list item is also identified as symmetric (S), as a pseudo-quandle (PQ), or a double pseudo-quandle (DPQ) wherever applicable. The two column invariants c_1 (the sum of the number of constant columns in U and D) and c_2 (the absolute difference of the number of constant columns in U and D are also shown.

- 3 biracks of size 2 (includes quandles, biquandles and racks)
- 2 biquandle of size 2 (includes quandles)
- 2 racks of size 2 (includes quandles)
- 1 quandle of size 2
- There are no biracks of size 2 that are not biquandles and not racks
- 1 biquandle of size 2 that is not a quandle
- 1 rack of size 2 that is not a quandle
- 2 quandle-related biracks of size 2
- 1 quandle-related birack of size 2 that is not a quandle
- There are no quandle-related biquandles of size 2 that are not quandles
- There are no essential virtual pairs of size 2
- There are no essential welded pairs of size 2
- 2 virtual invariant pairs of size 2 that are not essential virtual pairs or essential welded pairs
- 16 biracks of size 3 (includes quandles, biquandles and racks)
- 10 biquandles of size 3 (includes quandles)
- 6 racks of size 3 (includes quandles)
- 3 quandles of size 3
- 3 biracks of size 3 that are not biquandles and not racks
- 7 biquandles of size 3 that are not quandles
- 3 racks of size 3 that are not quandles
- 9 quandle-related biracks of size 3
- 6 quandle-related biracks of size 3 that are not quandles
- 1 quandle-related biquandle of size 3 that is not a quandle
- 3 essential virtual pairs of size 3
- 2 essential welded pairs of size 3
- 18 virtual invariant pairs of size 3 that are not essential virtual pairs or essential welded pairs
- 147 biracks of size 4 (includes quandles, biquandles and racks)
- 64 biquandle of size 4 (includes quandles)
- 19 racks of size 4 (includes quandles)
- 7 quandles of size 4
- 71 biracks of size 4 that are not biquandles and not racks
- 57 biquandles of size 4 that are not quandles
- 12 racks of size 4 that are not quandles
- 60 quandle-related biracks of size 4
- 53 quandle-related biracks of size 4 that are not quandles
- 11 quandle-related biquandles of size 4 that are not quandles
- 37 essential virtual pairs of size 4
- 8 essential welded pairs of size 4
- 298 virtual invariant pairs of size 4 that are not essential virtual pairs or essential welded pairs
- 703 quandle-related biracks of size 5 (includes quandles, biquandles and racks)
- 134 quandle-related biquandles of size 5 (includes quandles)
- 73 racks of size 5 (includes quandles)
- 21 quandles of size 5
- 517 quandle-related biracks of size 5 that are not biquandles and not racks
- 682 quandle-related biracks of size 5 that are not quandles (includes biquandles and racks)
- 113 quandle-related biquandles of size 5 that are not quandles
- 52 racks of size 5 that are not quandles
- 34 quandle-related essential virtual pairs of size 5
- 7 quandle-related essential welded pairs of size 5
- 471 quandle-related virtual invariant pairs of size 5 that are not essential virtual pairs or essential welded pairs
- 13562 quandle-related biracks of size 6 (includes quandles, biquandles and racks)
- 1578 quandle-related biquandles of size 6 (includes quandles)
- 352 racks of size 6 (includes quandles)
- 72 quandles of size 6
- 11704 quandle-related biracks of size 6 that are not biquandles and not racks
- 13490 quandle-related biracks of size 6 that are not quandles (includes biquandles and racks)
- 1506 quandle-related biquandles of size 6 that are not quandles
- 280 racks of size 6 that are not quandles
- 527 quandle-related essential virtual pairs of size 6
- 40 quandle-related essential welded pairs of size 6
- 9846 quandle-related virtual invariant pairs of size 6 that are not essential virtual pairs or essential welded pairs

The above lists were calculated from the plain format results using the following bash scripts

The script `convert-lists` converts the lists corresponding to a specified number of crossings into a specified format, the above lists were produced using commands `convert-lists <num-crossings> -S -o PT`, where `<num-crossings>` took values 2,...,6, `S` indicates that the source lists contain switches rather than sideways maps and `-o PT` identifies that the output should be in TeX format with rows written as permutations.

The script `redo-dominant-pairs` calculates the dominant pairs for crossings 2,...,6 and takes as a parameter the output format required. The above lists were produced using the command `redo-dominant-pairs -n 6 -TP`, where `-n 6` indicates that all lists up to order 6 should be created and `-TP` identifies TeX format with rows written as permutations.

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