Ian Wanless - cp's OEIS frontend

A378856 Minimum over groups of order n of the maximum order of an element of the group.

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 15, 2, 17, 3, 19, 5, 7, 11, 23, 4, 5, 13, 3, 14, 29, 15, 31, 2, 33, 17, 35, 4, 37, 19, 13, 10, 41, 7, 43, 22, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 21, 2, 65, 33, 67, 17, 69, 35, 71, 4, 73, 37, 5, 38, 77, 13, 79, 5, 3, 41, 83, 14, 85, 43, 87, 22, 89, 15, 91, 46, 31, 47, 95, 4, 97, 7, 33, 5

Offset: 1

Ian Wanless, Feb 10 2025

nonn

			When n is a power of a prime p, a(n) = p because all elements of the elementary abelian p-group have order 1 or p.

GAP small group library, The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.14.0; 2024. (https://www.gap-system.org).

a(n) is bounded below by sequence A006530 and above by A007947.

A350026 Number of species containing totally symmetric Latin squares of order n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 13, 8, 139, 65, 25888, 24316, 92798256, 122859796

Offset: 1

Ian Wanless, Dec 08 2021

A Latin square is "totally symmetric" if all 6 of its conjugates are equal. Species are also known as "main classes" or "paratopism classes".

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076019, A350025.

A350025 Number of totally symmetric Latin squares of order n.

Original entry on oeis.org

1, 2, 3, 16, 30, 480, 1290, 163200, 471240, 386400000, 2269270080, 12238171545600, 149648961369600, 8089070513113497600, 160650421233958656000, 91361407076595590705971200

Offset: 1

Ian Wanless, Dec 08 2021

A Latin square is "totally symmetric" if all 6 of its conjugates are equal.

Hy Ginsberg, Totally Symmetric Quasigroups of Order 16, arXiv preprint (2022). arXiv:2211.13204 [math.CO]
Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076019, A350026.

a(16) from Ginsberg link via Charles R Greathouse IV, Dec 02 2022

A350024 Number of diagonal semisymmetric quasigroups of order n.

Original entry on oeis.org

1, 0, 2, 1, 1, 0, 7, 2, 112, 2369, 347299, 237570420

Offset: 1

Ian Wanless, Dec 08 2021

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076021, A350022, A350023.

A350023 Number of diagonal semisymmetric Latin squares of order n.

Original entry on oeis.org

1, 0, 3, 2, 30, 0, 3000, 20160, 19571328, 8136806400, 13826847640320, 113788019281305600

Offset: 1

Ian Wanless, Dec 08 2021

"Diagonal" means that the symbols on the main diagonal must all be different.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A350022, A350024.

A350022 Number of idempotent semisymmetric Latin squares of order n.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 480, 0, 2274048, 757555200, 4693077997977600

Offset: 1

Ian Wanless, Dec 08 2021

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076021, A350023, A350024.

A350020 Number of species containing semisymmetric Latin squares of order n.

Original entry on oeis.org

1, 1, 1, 2, 2, 7, 28, 366, 13899, 1968997, 934327507

Offset: 1

Ian Wanless, Dec 08 2021

nonn

Species (also called "main classes") are the largest natural equivalence classes of Latin squares.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076016, A076017, A350019.

A350019 Number of isotopism classes containing semisymmetric Latin squares of order n.

Original entry on oeis.org

1, 1, 1, 2, 2, 7, 33, 557, 26511, 3908091, 1867909542

Offset: 1

Ian Wanless, Dec 08 2021

nonn

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A076016 A076017.

A350018 Number of unipotent commutative loops of order 2n.

Original entry on oeis.org

1, 1, 1, 7, 3460, 6320290037, 15859695832489637513

Offset: 1

Ian Wanless, Dec 08 2021

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

Cf. A089925, A350017.

A350017 Number of isotopism classes containing symmetric unipotent reduced Latin squares of order 2n.

Original entry on oeis.org

1, 1, 1, 6, 396, 526915616, 1132835421602062347

Offset: 1

Ian Wanless, Dec 08 2021

Isotopism classes are obtained by permuting rows, permuting columns and permuting symbols. There is a stronger notion of equivalence called "species" (also known as main classes and paratopism classes). For this particular problem the counts for species equal the counts for isotopism classes.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.

For odd n the terms equal A000474.

Cf. A350009.

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User: Ian Wanless

A378856 Minimum over groups of order n of the maximum order of an element of the group.

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A350026 Number of species containing totally symmetric Latin squares of order n.

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A350025 Number of totally symmetric Latin squares of order n.

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A350024 Number of diagonal semisymmetric quasigroups of order n.

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A350023 Number of diagonal semisymmetric Latin squares of order n.

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A350022 Number of idempotent semisymmetric Latin squares of order n.

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A350020 Number of species containing semisymmetric Latin squares of order n.

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A350019 Number of isotopism classes containing semisymmetric Latin squares of order n.

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A350018 Number of unipotent commutative loops of order 2n.

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A350017 Number of isotopism classes containing symmetric unipotent reduced Latin squares of order 2n.

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