A000438 -id:A000438 - cp's OEIS frontend

A035481 Number of n X n symmetric matrices whose first row is 1..n and whose rows and columns are all permutations of 1..n.

1, 1, 1, 1, 4, 6, 456, 6240, 10936320, 1225566720, 130025295912960, 252282619805368320, 2209617218725251404267520, 98758655816833727741338583040

Offset: 0

Joshua Zucker and Joe Keane

The odd subsequence is A000438. The even subsequence is A035483.

			a(3) = 1 because after 123 in the first row and column, 213 is not allowed for the second row, so it must be 231 and thus the third row is 312.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130, also on Enumeration of Latin squares with conjugate symmetry, arXiv:2104.07902 [math.CO], 2021. Table 2 p. 7.

Cf. A000438, A035482, A000315, A002860, A003090, A040082.

(* This script is not suitable for n > 6 *) matrices[n_ /; n > 1] := Module[{a, t, vars}, t = Table[Which[i==1, j, j==1, i, j>i, a[i, j], True, a[j, i]], {i, n}, {j, n}]; vars = Select[Flatten[t], !IntegerQ[#]& ] // Union; t /. {Reduce[And @@ (1 <= # <= n & /@ vars) && And @@ Unequal @@@ t, vars, Integers] // ToRules}]; a[0] = a[1] = 1; a[n_] := Length[ matrices[n]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 6}] (* Jean-François Alcover, Jan 04 2016 *)

a(10)-a(13) from Ian Wanless, Oct 20 2019

A003191 Number of symmetric Latin squares of order 2n with constant diagonal.

Original entry on oeis.org

1, 1, 6, 5972, 1225533120

Offset: 1

N. J. A. Sloane

N. T. Gridgeman, Latin squares under restriction and a jumboization, J. Rec. Math., 5 (No. 3, 1972), 198-202.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Index entries for sequences related to Latin squares and rectangles

Daniele Degiorgi (degiorgi(AT)inf.ethz.ch) suggests that this is an erroneous version of A000438.

A036981 Number of (2n+1) X (2n+1) symmetric matrices each of whose rows is a permutation of 1..(2n+1).

Original entry on oeis.org

1, 6, 720, 31449600, 444733651353600, 10070314878246926155776000, 614972203951464612786852376432607232000

Offset: 0

Joshua Zucker and Joe Keane

Number of different schedules for 2n+2 teams. - Andres Cardemil (andrescarde(AT)yahoo.com), Nov 28 2001

Minjia Shi, Li Xu, and Denis S. Krotov, The number of the non-full-rank Steiner triple systems, arXiv:1806.00009 [math.CO], 2018-2021.
Index entries for sequences related to tournaments.

Cf. A000438, A000474, A035481, A036980, A064120, A065594.

a(n) = A000438(n+1) * (2*n+1)!.

a(5)-a(6) computed from A000438 by Max Alekseyev, Jun 17 2011

A000474 Number of nonisomorphic 1-factorizations of complete graph K_{2n}.

Original entry on oeis.org

1, 1, 1, 6, 396, 526915620, 1132835421602062347

Offset: 1

N. J. A. Sloane

Number of essentially different ways of scheduling a tournament of 2n teams.

CRC Handbook of Combinatorial Designs (see pages 655, 720-723).
Jeffrey H. Dinitz, David K. Garnick, Brendan D. McKay, There are 526,915,620 nonisomorphic one-factorizations of K_{12}. J. Combin. Des. 2 (1994), no. 4, 273-285.
Petteri Kaski and Patric R. J. Östergård, There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations of K_{14}, Journal of Combinatorial Designs 17 (2009), pp. 147-159.
Charles C. Lindner, Eric Mendelsohn, and Alexander Rosa. "On the number of 1-factorizations of the complete graph." Journal of Combinatorial Theory, Series B 20.3 (1976): 265-282.
E. Seah and D. R. Stinson, On the enumeration of one-factorizations of complete graphs containing prescribed automorphism groups. Math. Comp. 50 (1988), 607-618.
W. D. Wallis, 1-Factorizations of complete graphs, pp. 593-631 in Jeffrey H. Dinitz and D. R. Stinson, Contemporary Design Theory, Wiley, 1992.

Petteri Kaski and Patric R. J. Östergård, There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations of K_{14}, arXiv:0801.0202 [math.CO], 2007.
Joseph Malkevitch, Mathematics and Sports
Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, arXiv:2104.07902 [math.CO], 2021. Table 5 p. 15.
D. V. Zinoviev, On the number of 1-factorizations of a complete graph [in Russian], Problemy Peredachi Informatsii, 50 (No. 4), 2014, 71-78.
Index entries for sequences related to tournaments

For odd n this sequence equals A350017. Cf. A000438.

a(n) ~ exp(2n^2 log(2n)) as n -> infinity (see CRC Handbook, p. 655, Theorem 4.20).

a(7) communicated by Vesa Linja-aho (vesa.linja-aho(AT)tkk.fi), Aug 02 2008

Comment, link, and update by Charles R Greathouse IV, May 11 2010

A035483 Number of 2n X 2n symmetric matrices whose first row is 1..2n and whose rows and columns are all permutations of 1..2n.

Original entry on oeis.org

1, 1, 4, 456, 10936320, 130025295912960, 2209617218725251404267520

Offset: 0

Joshua Zucker and Joe Keane

Cf. A000438, A035481, A000315, A002860, A003090, A040082.

a(n) = A035481(2*n). - Max Alekseyev, Apr 23 2010

a(5)-a(6) from Ian Wanless, Oct 20 2019

A035482 Number of n X n symmetric matrices each of whose rows is a permutation of 1..n.

Original entry on oeis.org

1, 1, 2, 6, 96, 720, 328320, 31449600, 440952422400, 444733651353600, 471835793808949248000, 10070314878246926155776000, 1058410183156945383046388908032000, 614972203951464612786852376432607232000

Offset: 0

Joshua Zucker and Joe Keane

The even and odd subsequences are A036980, A036981.

			a(3) = 6 because the first row is arbitrary (say, 213) and the rest is then determined. By symmetry the second row has to be 132 or 123 but in order for the third row/column to work it has to be 132.

Cf. A000438, A035481, A000315, A002860, A003090, A040082.

a(n) = A035481(n) * n!. [From Max Alekseyev, Apr 23 2010]

a(10)-a(13) (using A035481) from Alois P. Heinz, May 05 2023

A120488 Number of nonisomorphic perfect 1-factorizations of complete graph K_{2n}.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 23, 3155

Offset: 1

N. J. A. Sloane, Jul 22 2006

CRC Handbook Combin. Designs, p. 664.
Barbara M. Maenhaut, Perfect 1-factorizations of complete and complete bipartite graphs, talk given at 31st Australasian Conf. Combin. Math and Combin. Computing, Alice Springs, 2006.
Petrenyuk, L. and Petrenyuk, A.; Intersection of Perfect One-Factorizations of Complete Graphs, Cybernetics 16 (1980), 6-9.
Wallis, W. D.; 1-Factorizations of complete graphs, pp. 593-631 in J. H. Dinitz and D R. Stinson, Contemporary Design Theory, Wiley, 1992.

Jeffrey H. Dinitz and David K. Garnick, There are 23 Nonisomorphic Perfect One-Factorizations of K_{14}, J. Combin. Des. 4 (1996), no. 1, 1-3.
Wallis, W. D.; One-factorization of a graph, in Encyclopaedia of Mathematics (ed. Hazewinkel, Michiel), Springer-Verlag, 2002.

Cf. A000438, A000474, A120489.

Additional references from David Garnick (dgarnick(AT)gmail.com), Jan 17 2007
Edited by N. J. A. Sloane at the suggestion of Ian Wanless, Apr 01 2008
a(8) from Ian Wanless, Oct 20 2019

A365520 Number of 1-factorizations of complete graph K_{2n} that all share one arbitrary pairing in common.

Original entry on oeis.org

1, 1, 2, 416, 11672064, 266965735243776, 9500592190171594780311552

Offset: 1

Brian Lathrop, Sep 08 2023

Consider a round robin tournament with 2n teams competing. For the first round, select a set of pairings arbitrarily. a(n) is the number of possible round robin tournaments after choosing one of those sets of pairings.
a(n) can be obtained by dividing A000438(n) by the product (2n-3)*(2n-5)*...*3*1 when n > 2. For n = 1 or n = 2, a(n) is simply 1, since there is only one way to set up those rounds with 2 or 4 teams.
To show how this works, consider the number of possible round robin tournaments given 8 teams (A000438(4) = 6240) identified by the letters A through H.
Each of these tournaments can be constructed from a set of matchups of this form:
   (AB)(XX)(XX)(XX)
   (AC)(YY)(YY)(YY)
   (AD)(YY)(YY)(YY)
   (AE)(YY)(YY)(YY)
   (AF)(YY)(YY)(YY)
   (AG)(YY)(YY)(YY)
   (AH)(YY)(YY)(YY)
Note that the 6240 tournaments can be divided evenly by the number of ways the "X" teams can be paired up to fill the arbitrary round. This is why A001147(3) = 15 is a factor, since there are 5 ways to pick the first pair, followed by 3 ways to pick the second pair.
For larger numbers of teams, there are more ways to pair up; e.g., for 10 teams, there will be a factor of 7*5*3, and for 12 teams, a factor of 9*7*5*3, and so on.

			For n = 3, given teams A through F (2n), the only two round robin tournaments that share the pairing (AB)(CD)(EF) are:
   (AB)(CD)(EF)
   (AC)(BE)(DF)
   (AD)(BF)(CE)
   (AE)(BD)(CF)
   (AF)(BC)(DE)
and
   (AB)(CD)(EF)
   (AC)(BF)(DE)
   (AD)(BE)(CF)
   (AE)(BC)(DF)
   (AF)(BD)(CE)
which agrees with a(3) = 2.

Cf. A000438, A001147.

a(n) = A000438(n)/A001147(n-1).

cp's OEIS Frontend

A035481 Number of n X n symmetric matrices whose first row is 1..n and whose rows and columns are all permutations of 1..n.

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A003191 Number of symmetric Latin squares of order 2n with constant diagonal.

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A036981 Number of (2n+1) X (2n+1) symmetric matrices each of whose rows is a permutation of 1..(2n+1).

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A000474 Number of nonisomorphic 1-factorizations of complete graph K_{2n}.

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A035483 Number of 2n X 2n symmetric matrices whose first row is 1..2n and whose rows and columns are all permutations of 1..2n.

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A035482 Number of n X n symmetric matrices each of whose rows is a permutation of 1..n.

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A120488 Number of nonisomorphic perfect 1-factorizations of complete graph K_{2n}.

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A365520 Number of 1-factorizations of complete graph K_{2n} that all share one arbitrary pairing in common.

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