Michael Steyer - cp's OEIS frontend

A274332 Team size n for which there exists a balanced tournament for 2n+1 players so that in 2n+1 matches each player plays exactly n-1 times with and n times against each other player.

1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33

Offset: 0

Michael Steyer, Jun 22 2016

There are 2n+1 players and 2n+1 matches. In each match one person rests, and the remaining 2n players are divided into two equal teams.
Up to n=33 there is probably only a unique design (up to permutation), and it has point / mirror symmetry.
It is conjectured that this sequence is identical to A005097 (ref. Kohen link).
a(n) = A130290(n+2) = A102781(n+2) = A139791(n+1) = A005097(n+1) for 0 <= n <= 17. - Georg Fischer, Oct 30 2018

			n=5:
Match 1: 1,2,3,5,8 versus 4,7,9,10,11
Match 2: 2,3,4,6,9 versus 5,8,10,11,1
Matches 3..11: further cyclic permutations

Daniel Kohen and Ivan Sadofschi, A New Approach on the Seating Couples Problem, arXiv:1006.2571 [math.CO], 2010.

Cf. A005097, A102781, A130290, A139791.

Conjectured design scheme for Team 1 (N:= 2n+1; here players count from 0..2n): X, X+1 (mod N), X+1+2 (mod N), X+1+2+3 (mod N), ...; X = 0..2n (match number). Resting player: (X + (n*(n+1)/2) (mod N).

A181344 Number of n X n matrices over {0,1} with rows and columns summing to 3, rows and columns sorted (>=) by value.

Original entry on oeis.org

0, 0, 1, 1, 5, 25, 161, 1112, 8787, 76156, 728699, 7609065, 86162795, 1050755884, 13728407061, 191309852944

Offset: 1

Michael Steyer (m.steyer(AT)osram.de), Oct 14 2010

			n=4: {1110,1101,1011,0111} is the only matrix where each row (column) - read as a binary number - is equal to or larger than the previous one, so a(4)=1.

Bert Dobbelaere, Illustration of initial terms

Cf. A000512, A001501, A181345.

a(10)-a(16) from Bert Dobbelaere, Feb 23 2020

A181345 Number of n X n matrices over {0,1} with rows and columns summing to 3, rows and columns sorted (>) by value.

Original entry on oeis.org

0, 0, 0, 1, 2, 12, 87, 662, 5611, 51141, 509277, 5504398, 64122940, 800741192, 10673478573, 151323048909

Offset: 1

Michael Steyer (m.steyer(AT)osram.de), Oct 18 2010

			n=4: {1110,1101,1011,0111} is the only matrix where each row (column) - read as a binary number - is larger than the previous one, so a(4)=1.

Bert Dobbelaere, Illustration of initial terms

Cf. A000512, A001501, A181344.

Offset corrected and a(10)-a(16) from Bert Dobbelaere, Feb 23 2020

A079815 Number of equivalent classes of n X n 0-1 matrices with 3 1's in each row and column.

Original entry on oeis.org

0, 0, 1, 1, 2, 7, 16, 71

Offset: 1

Michael Steyer (m.steyer(AT)osram.de), Feb 20 2003

Matrices are considered to belong to the same equivalent class if they can be transformed into each other by successive permutations of rows or columns.
In general, to transform 2 equivalent matrices into each other, it is necessary to first permute rows, then columns, then rows and so on.
From Brendan McKay, Aug 27 2010: (Start)
A079815 appears on the surface to describe the same objects as A000512, but I don't know where the term "71" comes from.
Also the comment "In general, to transform 2 equivalent matrices into each other, it is necessary to first permute rows, then columns, then rows and so on." is wrong - actually only one permutation of rows and one permutation of columns is enough.
I will guess that this sequence counts matrices in which both the rows and columns are in sorted order. The reason I suspect that is because a common way to make such matrices is to alternately sort the rows and columns until it stabilizes.
The value of a(8) should be checked. (End)

			n=4: every matrix with 3 1's in each row and column can be transformed by permutation of rows (or columns) into {1110,1101,1011,0111}, therefore a(4)=1.

Cf. A001501.

Edited by N. J. A. Sloane, Sep 04 2010

A064852 Number of orbits in A002619 consisting of n permutations.

Original entry on oeis.org

1, 0, 0, 1, 4, 18, 102, 624, 4476, 36248, 329890, 3326054, 36846276, 444783906, 5811885808, 81729607680, 1230752346352, 19760412095328, 336967037143578, 6082255011151724, 115852476579789984, 2322315553090615850, 48869596859895986086, 1077167364116800207968

Offset: 1

Michael Steyer (m.steyer(AT)osram.de), Oct 06 2001

Also, the number of aperiodic oriented cycles on n nodes up to rotation of the nodes. See A324513 for illustrations of aperiodic undirected cycles. - Andrew Howroyd, Aug 16 2019

			n=6: The orbit {(124635)(235146)(346251)(451362)(562413)(613524)} consists of 6 single permutations.

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A000010, A002619, A008683, A047918, A324513.

a[n_] := Sum[ MoebiusMu[n/k] * EulerPhi[n/k] * (n/k)^k * (k!/n^2), {k, Divisors[n]}]; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Jun 26 2012, after PARI *)

for(n=1,23,print(sumdiv(n,d,moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2)))

{ for (n=1, 100, a=sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2); write("b064852.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 28 2009

a(n) = Sum_{k|n} mu(n/k)*phi(n/k)*(n/k)^k*k!/n^2 = A047918(n, n)/n^2.

Corrected and extended by Jason Earls and Vladeta Jovovic, Oct 08 2001

A058844 Number of ways of placing n labeled balls into 4 indistinguishable boxes with at least 2 balls in each box.

Original entry on oeis.org

105, 1260, 9450, 56980, 302995, 1487200, 6914908, 30950920, 134779645, 575156036, 2417578670, 10046531276, 41388056231, 169371383384, 689568172832, 2796362035104, 11305163394129, 45595968007260, 183557935474290, 737897437077060, 2963015460969915

Offset: 8

Michael Steyer, Dec 02 2000

			a(8) = 8!/(2!*2!*2!*2!*4!) = 105.

T. D. Noe, Table of n, a(n) for n=8..200
Daniel J. Bernstein and Andreas Hülsing, Decisional second-preimage resistance: When does SPR imply PRE?, (2019).
Erik Vigren and Andreas Dieckmann, A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.

Cf. A000247 (2 boxes), A000478 (3 boxes).

[(4^n-3^(n-1)*(4*n+12)+2^(n-1)*(12+9*n+3*n^2)-4*n^3-8*n-4)/24 : n in [8..25]]; // Wesley Ivan Hurt, Oct 28 2014

A058844:=n->(4^n-3^(n-1)*(4*n+12)+2^(n-1)*(12+9*n+3*n^2)-4*n^3-8*n-4)/24: seq(A058844(n), n=8..25); # Wesley Ivan Hurt, Oct 28 2014

Table[(4^n - 3^(n - 1) (4 n + 12) + 2^(n - 1) (12 + 9 n + 3 n^2) - 4 n^3 - 8 n - 4)/24, {n, 8, 25}] (* Wesley Ivan Hurt, Oct 28 2014 *)
offset = 8; terms = 21; egf = (Exp[x]-1-x)^4/4!; Drop[CoefficientList[egf + O[x]^(terms+offset), x]*Range[0, terms+offset-1]!, offset] (* Jean-François Alcover, May 07 2017 *)

a(n)=(4^n - 3^(n-1)*(4*n+12) + 2^(n-1)*(12+9*n+3*n^2) - 4*n^3 - 8*n - 4)/24 \\ Charles R Greathouse IV, Oct 28 2014

E.g.f.: ((exp(x) - 1 - x)^4)/4!.
G.f.: x^8*(288*x^6 - 1560*x^5 + 3500*x^4 - 4130*x^3 + 2625*x^2 - 840*x + 105) / ((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^2*(1 - 4*x)).
a(n) = (4^n-3^(n-1)(4n+12)+2^(n-1)(12+9n+3n^2)-4n^3-8n-4)/24. - David Wasserman, Jun 06 2007

More terms from James Sellers, Dec 06 2000

cp's OEIS Frontend

User: Michael Steyer

A274332 Team size n for which there exists a balanced tournament for 2n+1 players so that in 2n+1 matches each player plays exactly n-1 times with and n times against each other player.

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A181344 Number of n X n matrices over {0,1} with rows and columns summing to 3, rows and columns sorted (>=) by value.

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A181345 Number of n X n matrices over {0,1} with rows and columns summing to 3, rows and columns sorted (>) by value.

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Examples

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Extensions

A079815 Number of equivalent classes of n X n 0-1 matrices with 3 1's in each row and column.

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A064852 Number of orbits in A002619 consisting of n permutations.

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A058844 Number of ways of placing n labeled balls into 4 indistinguishable boxes with at least 2 balls in each box.

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