pith. machine review for the scientific record. sign in

arxiv: 2605.12859 · v1 · submitted 2026-05-13 · 🧮 math.CO

Recognition: no theorem link

A study on Type-2 isomorphic circulant graphs. Part 4: 960 triples of Type-2 isomorphic circulant graphs C₅₄(R)

Vilfred Kamalappan
Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:59 UTC · model grok-4.3

classification 🧮 math.CO
keywords circulant graphsType-2 isomorphismisomorphic triplesorder 54connection setsgraph enumeration
0
0 comments X

The pith

There are 960 triples of Type-2 isomorphic circulant graphs of order 54, each with respect to m=3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper is the fourth part of a series that classifies circulant graphs under Type-2 isomorphism. It focuses on graphs with 54 vertices and counts the triples that are isomorphic under this relation. Enumeration of the possible connection sets produces exactly 960 such triples. Every identified triple satisfies the Type-2 condition specifically when the parameter m is set to 3. The result supplies a concrete count that later parts of the series can use for comparison with other orders.

Core claim

The paper shows that there are 960 triples of Type-2 isomorphic circulant graphs C_54(R) and that each triple is of Type-2 isomorphic with respect to m = 3.

What carries the argument

Enumeration of connection sets R to locate triples of circulant graphs C_54(R) that are Type-2 isomorphic.

Load-bearing premise

The enumeration procedure from the prior parts correctly and exhaustively identifies all Type-2 isomorphic triples for order 54 without omissions or duplicates.

What would settle it

An independent search that returns a total different from 960, or that finds even one triple that is Type-2 isomorphic but not with respect to m=3, would disprove the claim.

read the original abstract

This study is the $4^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10} and is a continuation of Part 3. Here, we study Type-2 isomorphic circulant graphs of order 54 and show that there are 960 triples of Type-2 isomorphic circulant graphs of order 54 and each triple of isomorphic circulant graphs is of Type-2 isomorphic w.r.t. $m$ = 3.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper is the fourth part of a ten-part series on Type-2 isomorphic circulant graphs. It enumerates circulant graphs C_54(R) and claims there exist exactly 960 triples of Type-2 isomorphic graphs of order 54, with each triple being Type-2 isomorphic with respect to parameter m=3.

Significance. If the enumeration is complete and free of overcounting or omissions, the result supplies a precise count for a specific order in the ongoing classification of circulant graph isomorphisms. Such enumerative data can support broader studies of graph symmetries, with potential relevance to combinatorial designs or network analysis. The series format allows incremental accumulation of results, but the absence of verification artifacts in this installment renders the significance provisional pending independent checks.

major comments (1)
  1. [Main body (enumeration section)] The central claim of exactly 960 triples rests on the enumeration procedure from parts 1-3, yet the manuscript neither reproduces the algorithm, lists the 960 triples, nor supplies computational output, code, or error bounds specific to n=54. This omission is load-bearing because the count cannot be verified or falsified from the text alone.
minor comments (1)
  1. [Abstract] The abstract states the final count but omits any reference to the concrete method or prior-part algorithm; adding one sentence on the enumeration approach would improve self-containment.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address the major comment regarding the enumeration procedure and verifiability below, clarifying the role of the series while agreeing to enhancements for improved accessibility.

read point-by-point responses

  1. Referee: The central claim of exactly 960 triples rests on the enumeration procedure from parts 1-3, yet the manuscript neither reproduces the algorithm, lists the 960 triples, nor supplies computational output, code, or error bounds specific to n=54. This omission is load-bearing because the count cannot be verified or falsified from the text alone.

    Authors: The enumeration algorithm and its theoretical basis are fully developed and presented in Parts 1-3 of the series, to which this manuscript is a direct continuation. For order 54 we apply this established procedure exhaustively to identify all Type-2 isomorphic triples with parameter m=3, obtaining the count of 960. In the revised version we will add a concise, self-contained summary of the algorithm steps as applied to n=54, including the relevant search parameters and decision criteria. We will also include high-level pseudocode describing the computational implementation. The complete list of 960 triples is too voluminous for the main text; we will instead supply it as supplementary material. Because the method is a deterministic, exhaustive enumeration of a finite set, probabilistic error bounds are not applicable, but we will add an explicit statement confirming that the search covers the entire space of possible connection sets for this order. These changes will allow independent verification without requiring the reader to consult every prior part. revision: partial

Circularity Check

0 steps flagged

Enumeration count presented as output of prior procedure with no definitional reduction

full rationale

The paper states its central claim as the result of applying an enumeration procedure developed across a cited series of prior parts. No equations, parameters, or derivations appear in the abstract or described text that would make the reported total of 960 triples equivalent to its inputs by construction. The self-citations describe the method used but do not invoke a uniqueness theorem, ansatz, or fitted value that is then renamed as a prediction. The claim remains an externally falsifiable enumeration result rather than a tautological restatement of prior content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of circulant graphs and Type-2 isomorphism carried forward from the earlier parts of the series; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard definitions of circulant graphs and Type-2 isomorphism from prior parts of the series
    The abstract invokes these definitions without re-deriving them.

pith-pipeline@v0.9.0 · 5391 in / 1181 out tokens · 29952 ms · 2026-05-14T18:59:15.212882+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

300 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    Adam,Research problem 2-10, J

    A. Adam,Research problem 2-10, J. Combinatorial Theory,3(1967), 393

    1967

  2. [2]

    A study on Type-2 isomorphic circulant graphs. Part 2: Type-2 isomorphic circulant graphs of orders 16, 24, 27

    V. Vilfred Kamalappan,All Type-2 Isomorphic Circulant GraphsC 16(R)andC 24(S), arXiv: 2508.09384v1 [math.CO] 12 Aug 2025, 28 pages

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  3. [3]

    Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs and related abelian groups, arXiv ID: 2012.11372v11 [math.CO] (26 Nov 2024)

    V. Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs and related Abelian Groups, arXiv: 2012.11372v11 [math.CO] (26 Nov. 2024), 183 pages

    work page arXiv 2012

  4. [4]

    Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs

    V. Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs. Part 1: Type-2 isomorphic circulant graphsC n(R)w.r.t.m= 2. Preprint. 31 pages

  5. [5]

    Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs

    V. Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs. Part 2: Type-2 isomorphic circulant graphs of orders 16, 24, 27. Preprint. 32 pages

  6. [6]

    Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs

    V. Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs. Part 3: 384 pairs of Type-2 isomorphic circulant graphsC 32(R). Preprint. 42 pages

  7. [7]

    Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs

    V. Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs. Part 4: 960 triples of Type-2 isomorphic circulant graphsC 54(R). Preprint. 76 pages

  8. [8]

    Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs

    V. Vilfred Kamalappan,A study on Type-2 isomorphic circulant graphs. Part 5: Type-2 isomorphic circulant graphs of orders 48, 81, 96. Preprint. 33 pages

  9. [9]

    Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs

    V. Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs. Part 6: Abelian groups(T2 n,m(Cn(R)),◦)and(V n,m(Cn(R)),◦). Preprint. 19 pages

  10. [10]

    Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs

    V. Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs. Part 7: Isomorphism series, digraph and graph ofC n(R). Preprint. 54 pages A STUDY ON TYPE-2 ISOMORPHICC n(R): PART 4: 960 TRIPLES OF TYPE-2 ISOMORPHICC 54(R) 31 Table 1.Findingθ 54,3,2(C54(R)),θ54,3,4(C54(R)) andT1 54(C54(R)) asθ54,3,6(C54(R)) =C 54(R). S. No. C54(R) θ54,3,2(C54(R)) = θ...

  11. [11]

    C54(1,3,17,19) C54(3,7,11,25) C54(3,5,13,23) C54(1,3,17,19),C54(5,13,15,23), C54(7,11,21,25). T2

  12. [12]

    C54(1,6,17,19) C54(6,7,11,25) C54(5,6,13,23) C54(1,6,17,19),C54(5,13,23,24), C54(7,11,12,25). T2

  13. [13]

    C54(1,9,17,19) C54(7,9,11,25) C54(5,9,13,23) C54(1,9,17,19),C54(5,9,13,23), C54(7,9,11,25). T1

  14. [14]

    C54(1,12,17,19) C54(7,11,12,25) C54(5,12,13,23) C54(1,12,17,19),C54(5,6,13,23), C54(7,11,24,25). T2

  15. [15]

    C54(1,15,17,19) C54(7,11,15,25) C54(5,13,15,23) C54(1,15,17,19),C54(5,13,21,23), C54(3,7,11,25). T2

  16. [16]

    C54(1,17,18,19) C54(7,11,18,25) C54(5,13,18,23) C54(1,17,18,19),C54(5,13,18,23), C54(7,11,18,25). T1

  17. [17]

    C54(1,17,19,21) C54(7,11,21,25) C54(5,13,21,23) C54(1,17,19,21),C54(3,5,13,23), C54(7,11,15,25). T2

  18. [18]

    C54(1,17,19,24) C54(7,11,24,25) C54(5,13,23,24) C54(1,17,19,24),C54(5,12,13,23), C54(6,7,11,25). T2

  19. [19]

    C54(1,17,19,27) C54(7,11,25,27) C54(5,13,23,27) C54(1,17,19,27),C54(5,13,23,27), C54(7,11,25,27). T1

  20. [20]

    C54(1,3,6,17,19) C54(3,6,7,11,25) C54(3,5,6,13,23) C54(1,3,6,17,19),C54(5,13,15,23,24), C54(7,11,12,21,25). T2

  21. [21]

    C54(1,3,9,17,19) C54(3,7,9,11,25) C54(3,5,9,13,23) C54(1,3,9,17,19),C54(5,9,13,15,23), C54(7,9,11,21,25). T2

  22. [22]

    C54(1,3,12,17,19) C54(3,7,11,12,25) C54(3,5,12,13,23) C54(1,3,12,17,19),C54(5,6,13,15,23), C54(7,11,21,24,25). T2

  23. [23]

    C54(1,3,15,17,19) C54(3,7,11,15,25) C54(3,5,13,15,23)C54(1,3,15,17,19),C54(5,13,15,21,23), C54(3,7,11,21,25). T2

  24. [24]

    C54(1,3,17,18,19) C54(3,7,11,18,25) C54(3,5,13,18,23)C54(1,3,17,18,19),C54(5,13,15,18,23), C54(7,11,18,21,25). T2

  25. [25]

    C54(1,3,17,19,21) C54(3,7,11,21,25) C54(3,5,13,21,23) C54(1,3,17,19,21),C54(3,5,13,15,23), C54(7,11,15,21,25). T2

  26. [26]

    C54(1,3,17,19,24) C54(3,7,11,24,25) C54(3,5,13,23,24)C54(1,3,17,19,24),C54(5,12,13,15,23), C54(6,7,11,21,25). T2

  27. [27]

    C54(1,3,17,19,27) C54(3,7,11,25,27) C54(3,5,13,23,27)C54(1,3,17,19,27),C54(5,13,15,23,27), C54(7,11,21,25,27). T2

  28. [28]

    C54(1,6,9,17,19) C54(6,7,9,11,25) C54(5,6,9,13,23) C54(1,6,9,17,19),C54(5,9,13,23,24), C54(7,9,11,12,25). T2

  29. [29]

    C54(1,6,12,17,19) C54(6,7,11,12,25) C54(5,6,12,13,23) C54(1,6,12,17,19),C54(5,6,13,23,24), C54(7,11,12,24,25). T2

  30. [30]

    C54(1,6,15,17,19) C54(6,7,11,15,25) C54(5,6,13,15,23)C54(1,6,15,17,19),C54(5,13,21,23,24), C54(3,7,11,12,25). T2

  31. [31]

    C54(1,6,17,18,19) C54(6,7,11,18,25) C54(5,6,13,18,23)C54(1,6,17,18,19),C54(5,13,18,23,24), C54(7,11,12,18,25). T2

  32. [32]

    C54(1,6,17,19,21) C54(6,7,11,21,25) C54(5,6,13,21,23) C54(1,6,17,19,21),C54(3,5,13,23,24), C54(7,11,12,15,25). T2

  33. [33]

    C54(1,6,17,19,24) C54(6,7,11,24,25) C54(5,6,13,23,24)C54(1,6,17,19,24),C54(5,12,13,23,24), C54(6,7,11,12,25). T2

  34. [34]

    C54(1,6,17,19,27) C54(6,7,11,25,27) C54(5,6,13,23,27)C54(1,6,17,19,27),C54(5,13,23,24,27), C54(7,11,12,25,27). T2

  35. [35]

    C54(1,9,12,17,19) C54(7,9,11,12,25) C54(5,9,12,13,23) C54(1,9,12,17,19),C54(5,6,9,13,23), C54(7,9,11,24,25). T2

  36. [36]

    C54(1,9,15,17,19) C54(7,9,11,15,25) C54(5,9,13,15,23) C54(1,9,15,17,19),C54(5,9,13,21,23), C54(3,7,9,11,25). T2

  37. [37]

    C54(1,9,17,18,19) C54(7,9,11,18,25) C54(5,9,13,18,23) C54(1,9,17,18,19),C54(5,9,13,18,23), C54(7,9,11,18,25). T1

  38. [38]

    C54(1,9,17,19,21) C54(7,9,11,21,25) C54(5,9,13,21,23) C54(1,9,17,19,21),C54(3,5,9,13,23), C54(7,9,11,15,25). T2

  39. [39]

    C54(1,9,17,19,24) C54(7,9,11,24,25) C54(5,9,13,23,24) C54(1,9,17,19,24),C54(5,9,12,13,23), C54(6,7,9,11,25). T2

  40. [40]

    C54(1,9,17,19,27) C54(7,9,11,25,27) C54(5,9,13,23,27) C54(1,9,17,19,27),C54(5,9,13,23,27), C54(7,9,11,25,27). T1

  41. [41]

    T2 32VILFRED KAMALAPPAN Table 2.Findingθ 54,3,2(C54(R)),θ54,3,4(C54(R)) andT1 54(C54(R)) asθ54,3,6(C54(R)) =C 54(R)

    C54(1,12,15,17,19)C54(7,11,12,15,25)C54(5,12,13,15,23)C54(1,12,15,17,19),C54(5,6,13,21,23), C54(3,7,11,24,25). T2 32VILFRED KAMALAPPAN Table 2.Findingθ 54,3,2(C54(R)),θ54,3,4(C54(R)) andT1 54(C54(R)) asθ54,3,6(C54(R)) =C 54(R). S. No. C54(R) θ54,3,2(C54(R)) = θ54,3,4(C54(R)) = C54(S)∈T154(C54(R)) T1 or T2

  42. [42]

    C54(1,12,17,18,19)C54(7,11,12,18,25)C54(5,12,13,18,23)C54(1,12,17,18,19),C54(5,6,13,18,23), C54(7,11,18,24,25). T2

  43. [43]

    C54(1,12,17,19,21)C54(7,11,12,21,25)C54(5,12,13,21,23) C54(1,12,17,19,21),C54(3,5,6,13,23), C54(7,11,15,24,25). T2

  44. [44]

    C54(1,12,17,19,24)C54(7,11,12,24,25)C54(5,12,13,23,24)C54(1,12,17,19,24),C54(5,6,12,13,23), C54(6,7,11,24,25). T2

  45. [45]

    C54(1,12,17,19,27)C54(7,11,12,25,27)C54(5,12,13,23,27)C54(1,12,17,19,27),C54(5,6,13,23,27), C54(7,11,24,25,27). T2

  46. [46]

    C54(1,15,17,18,19)C54(7,11,15,18,25)C54(5,13,15,18,23)C54(1,15,17,18,19),C54(5,13,18,21,23), C54(3,7,11,18,25). T2

  47. [47]

    C54(1,15,17,19,21)C54(7,11,15,21,25)C54(5,13,15,21,23)C54(1,15,17,19,21),C54(3,5,13,21,23), C54(3,7,11,15,25). T2

  48. [48]

    C54(1,15,17,19,24)C54(7,11,15,24,25)C54(5,13,15,23,24)C54(1,15,17,19,24),C54(5,12,13,21,23), C54(3,6,7,11,25). T2

  49. [49]

    C54(1,15,17,19,27)C54(7,11,15,25,27)C54(5,13,15,23,27)C54(1,15,17,19,27),C54(5,13,21,23,27), C54(3,7,11,25,27). T2

  50. [50]

    C54(1,17,18,19,21)C54(7,11,18,21,25)C54(5,13,18,21,23)C54(1,17,18,19,21),C54(3,5,13,18,23), C54(7,11,15,18,25). T2

  51. [51]

    C54(1,17,18,19,24)C54(7,11,18,24,25)C54(5,13,18,23,24)C54(1,17,18,19,24),C54(5,12,13,18,23), C54(6,7,11,18,25). T2

  52. [52]

    C54(1,17,18,19,27)C54(7,11,18,25,27)C54(5,13,18,23,27)C54(1,17,18,19,27),C54(5,13,18,23,27), C54(7,11,18,25,27). T1

  53. [53]

    C54(1,17,19,21,24)C54(7,11,21,24,25)C54(5,13,21,23,24)C54(1,17,19,21,24),C54(3,5,12,13,23), C54(6,7,11,15,25). T2

  54. [54]

    C54(1,17,19,21,27)C54(7,11,21,25,27)C54(5,13,21,23,27)C54(1,17,19,21,27),C54(3,5,13,23,27), C54(7,11,15,25,27). T2

  55. [55]

    C54(1,17,19,24,27)C54(7,11,24,25,27)C54(5,13,23,24,27)C54(1,17,19,24,27),C54(5,12,13,23,27), C54(6,7,11,25,27). T2

  56. [56]

    C54(1,3,6,9,17,19) C54(3,6,7,9,11,25) C54(3,5,6,9,13,23) C54(1,3,6,9,17,19), C54(5,9,13,15,23,24), C54(7,9,11,12,21,25). T2

  57. [57]

    C54(1,3,6,12,17,19)C54(3,6,7,11,12,25)C54(3,5,6,12,13,23) C54(1,3,6,12,17,19), C54(5,6,13,15,23,24), C54(7,11,12,21,24,25). T2

  58. [58]

    C54(1,3,6,15,17,19)C54(3,6,7,11,15,25)C54(3,5,6,13,15,23) C54(1,3,6,15,17,19), C54(5,13,15,21,23,24), C54(3,7,11,12,21,25). T2

  59. [59]

    C54(1,3,6,17,18,19)C54(3,6,7,11,18,25)C54(3,5,6,13,18,23) C54(1,3,6,17,18,19), C54(5,13,15,18,23,24), C54(7,11,12,18,21,25). T2

  60. [60]

    C54(1,3,6,17,19,21)C54(3,6,7,11,21,25)C54(3,5,6,13,21,23) C54(1,3,6,17,19,21), C54(3,5,13,15,23,24), C54(7,11,12,15,21,25). T2

  61. [61]

    C54(1,3,6,17,19,24)C54(3,6,7,11,24,25)C54(3,5,6,13,23,24) C54(1,3,6,17,19,24), C54(5,12,13,15,23,24), C54(6,7,11,12,21,25). T2

  62. [62]

    C54(1,3,6,17,19,27)C54(3,6,7,11,25,27)C54(3,5,6,13,23,27) C54(1,3,6,17,19,27), C54(5,13,15,23,24,27), C54(7,11,12,21,25,27). T2

  63. [63]

    C54(1,3,9,12,17,19)C54(3,7,9,11,12,25)C54(3,5,9,12,13,23) C54(1,3,9,12,17,19), C54(5,6,9,13,15,23), C54(7,9,11,21,24,25). T2

  64. [64]

    C54(1,3,9,15,17,19)C54(3,7,9,11,15,25)C54(3,5,9,13,15,23) C54(1,3,9,15,17,19), C54(5,9,13,15,21,23), C54(3,7,9,11,21,25). T2

  65. [65]

    C54(1,3,9,17,18,19)C54(3,7,9,11,18,25)C54(3,5,9,13,18,23) C54(1,3,9,17,18,19), C54(5,9,13,15,18,23), C54(7,9,11,18,21,25). T2

  66. [66]

    C54(1,3,9,17,19,21)C54(3,7,9,11,21,25)C54(3,5,9,13,21,23) C54(1,3,9,17,19,21), C54(3,5,9,13,15,23), C54(7,9,11,15,21,25). T2 A STUDY ON TYPE-2 ISOMORPHICC n(R): PART 4: 960 TRIPLES OF TYPE-2 ISOMORPHICC 54(R) 33 Table 3.Findingθ 54,3,2(C54(R)),θ54,3,4(C54(R)) andT1 54(C54(R)) asθ54,3,6(C54(R)) =C 54(R). S. No. C54(R) θ54,3,2(C54(R)) = θ54,3,4(C54(R)) = C5...

  67. [67]

    C54(1,3,9,17,19,24) C54(3,7,9,11,24,25) C54(3,5,9,13,23,24) C54(1,3,9,17,19,24), C54(5,9,12,13,15,23), C54(6,7,9,11,21,25). T2

  68. [68]

    C54(1,3,9,17,19,27) C54(3,7,9,11,25,27) C54(3,5,9,13,23,27) C54(1,3,9,17,19,27), C54(5,9,13,15,23,27), C54(7,9,11,21,25,27). T2

  69. [69]

    C54(1,3,12,15,17,19)C54(3,7,11,12,15,25)C54(3,5,12,13,15,23)C54(1,3,12,15,17,19), C54(5,6,13,15,21,23), C54(3,7,11,21,24,25). T2

  70. [70]

    C54(1,3,12,17,18,19)C54(3,7,11,12,18,25)C54(3,5,12,13,18,23)C54(1,3,12,17,18,19), C54(5,6,13,15,18,23), C54(7,11,18,21,24,25). T2

  71. [71]

    C54(1,3,12,17,19,21)C54(3,7,11,12,21,25)C54(3,5,12,13,21,23)C54(1,3,12,17,19,21), C54(3,5,6,13,15,23), C54(7,11,15,21,24,25). T2

  72. [72]

    C54(1,3,12,17,19,24)C54(3,7,11,12,24,25)C54(3,5,12,13,23,24)C54(1,3,12,17,19,24), C54(5,6,12,13,15,23), C54(6,7,11,21,24,25). T2

  73. [73]

    C54(1,3,12,17,19,27)C54(3,7,11,12,25,27)C54(3,5,12,13,23,27)C54(1,3,12,17,19,27), C54(5,6,13,15,23,27), C54(7,11,21,24,25,27). T2

  74. [74]

    C54(1,3,15,17,18,19)C54(3,7,11,15,18,25)C54(3,5,13,15,18,23)C54(1,3,15,17,18,19), C54(5,13,15,18,21,23), C54(3,7,11,18,21,25). T2

  75. [75]

    C54(1,3,15,17,19,21)C54(3,7,11,15,21,25)C54(3,5,13,15,21,23)C54(1,3,15,17,19,21), C54(3,5,13,15,21,23), C54(3,7,11,15,21,25). T1

  76. [76]

    C54(1,3,15,17,19,24)C54(3,7,11,15,24,25)C54(3,5,13,15,23,24)C54(1,3,15,17,19,24), C54(5,12,13,15,21,23), C54(3,6,7,11,21,25). T2

  77. [77]

    C54(1,3,15,17,19,27)C54(3,7,11,15,25,27)C54(3,5,13,15,23,27)C54(1,3,15,17,19,27), C54(5,13,15,21,23,27), C54(3,7,11,21,25,27). T2

  78. [78]

    C54(1,3,17,18,19,21)C54(3,7,11,18,21,25)C54(3,5,13,18,21,23)C54(1,3,17,18,19,21), C54(3,5,13,15,18,23), C54(7,11,15,18,21,25). T2

  79. [79]

    C54(1,3,17,18,19,24)C54(3,7,11,18,24,25)C54(3,5,13,18,23,24)C54(1,3,17,18,19,24), C54(5,12,13,15,18,23), C54(6,7,11,18,21,25). T2

  80. [80]

    C54(1,3,17,18,19,27)C54(3,7,11,18,25,27)C54(3,5,13,18,23,27)C54(1,3,17,18,19,27), C54(5,13,15,18,23,27), C54(7,11,18,21,25,27). T2

Showing first 80 references.