A002848 -id:A002848 - cp's OEIS frontend

A152255 Erroneous version of A002848.

0, 0, 0, 1, 1, 2, 2, 3, 7, 15, 12, 30, 8, 32, 162, 21

Offset: 0

dead

R. K. Guy, ``Sedlacek's Conjecture on Disjoint Solutions of x+y= z,'' Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, ``Sedlacek's Conjecture on Disjoint Solutions of x+y= z,'' in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms.

1, 1, 2, 5, 15, 55, 232, 1161, 6643, 44566, 327064, 2709050, 24312028, 240833770, 2546215687, 29251369570, 355838858402, 4658866773664, 64127566159756, 940320691236206

Offset: 0

Jonas Wallgren, Mar 17 2005

			{{{1,2,3},{4,5,6},{7,8,9}}, {{1,2,3},{4,6,8},{5,7,9}}, {{1,3,5},{2,4,6},{7,8,9}}, {{1,4,7},{2,5,8},{3,6,9}}, {{1,5,9},{2,3,4},{6,7,8}}} are the 5 ways to split 1, 2, 3, ..., 9 into 3 arithmetic progressions each with 3 elements. Thus a(3)=5.

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]. See sequence "M".
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] Gives a(0)-a(10).

Cf. A104430-A104443.
All of A279197, A279198, A202705, A279199, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
See also A002848, A002849, A334250.

a(11)-a(14) from Alois P. Heinz, Dec 28 2011
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
a(18)-a(19) from Martin Fuller, Jul 08 2025

A002849 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n}, each satisfying X + Y = Z.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 3, 10, 25, 12, 42, 8, 40, 204, 21, 135, 1002, 4228, 720, 5134, 29546, 4079, 35533, 3040, 28777, 281504, 20505, 212283, 2352469, 16907265, 1669221, 19424213, 167977344, 14708525, 191825926, 10567748, 149151774, 2102286756, 103372655, 1534969405

Offset: 1

N. J. A. Sloane

nonn

			For n = 3, the unique solution is 1 + 2 = 3.
For n = 12, there are 8 solutions:
  1  5  6 | 1  5  6 | 2  5  7 | 1  6  7
  2  8 10 | 3  7 10 | 3  6  9 | 4  5  9
  4  7 11 | 2  9 11 | 1 10 11 | 3  8 11
  3  9 12 | 4  8 12 | 4  8 12 | 2 10 12
  --------+---------+---------+--------
  2  4  6 | 2  6  8 | 3  4  7 | 3  5  8
  1  9 10 | 4  5  9 | 1  8  9 | 2  7  9
  3  8 11 | 3  7 10 | 5  6 11 | 4  6 10
  5  7 12 | 1 11 12 | 2 10 12 | 1 11 12

R. K. Guy, "Sedlacek's Conjecture on Disjoint Solutions of x+y= z," in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, "Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics," in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Martin Fuller, Table of n, a(n) for n = 1..52 (terms 1..42 from Fausto A. C. Cariboni, terms 43..44 from Frank Niedermeyer)
R. K. Guy, Letter to N. J. A. Sloane, Jun 24 1971: front, back [Annotated scanned copy, with permission]
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971. [Annotated scanned copy, with permission]
Richard K. Guy, The unity of combinatorics, in Proc. 25th Iran. Math. Conf., Tehran, (1994), Math. Appl. 329 (1994) 129-159, Kluwer Acad. Publ., Dordrecht, 1995.
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.
Nigel Martin, Solving a conjecture of Sedlacek: maximal edge sets in the 3-uniform sumset hypergraphs, Discrete Mathematics, Volume 125, 1994, pp. 273-277.
Matheplanet Calculating sequence element a(16) of OEIS A108235
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Cf. A002848, A108235, A161826.

nxyz(v,t)=local(n,r,x2); r=0; if(t==0,return(1)); for(i3=3*t,#v, n=v[i3]; for(i1=1,i3-2, x2=n-v[i1]; if(x2<=v[i1],break); for(i2=i1+1,i3-1, if(v[i2]>=x2, if(v[i2]==x2, r+=nxyz(vector(i3-3,k,v[if(kFranklin T. Adams-Watters

Edited by N. J. A. Sloane, Feb 10 2010, based on posting to the Sequence Fans Mailing List by Franklin T. Adams-Watters, R. K. Guy, R. H. Hardin, Alois P. Heinz, Andrew Weimholt, Max Alekseyev and others
a(32)-a(39) from Max Alekseyev, Feb 23 2012
Definition corrected by Max Alekseyev, Nov 16 2012, Jul 06 2023
a(40)-a(41) from Fausto A. C. Cariboni, Feb 04 2017
a(42) from Fausto A. C. Cariboni, Mar 12 2017

A108235 Number of partitions of {1,2,...,3n} into n triples (X,Y,Z) each satisfying X+Y=Z.

Original entry on oeis.org

1, 1, 0, 0, 8, 21, 0, 0, 3040, 20505, 0, 0, 10567748, 103372655, 0, 0, 142664107305, 1836652173363, 0, 0

Offset: 0

N. J. A. Sloane, Feb 10 2010, based on posting to the Sequence Fans Mailing List by Franklin T. Adams-Watters, R. K. Guy, R. H. Hardin, Alois P. Heinz, Andrew Weimholt and others

a(0)=1 by convention.

			For m = 1 the unique solution is 1 + 2 = 3.
For m = 4 there are 8 solutions:
  1  5  6 | 1  5  6 | 2  5  7 | 1  6  7
  2  8 10 | 3  7 10 | 3  6  9 | 4  5  9
  4  7 11 | 2  9 11 | 1 10 11 | 3  8 11
  3  9 12 | 4  8 12 | 4  8 12 | 2 10 12
  --------+---------+---------+--------
  2  4  6 | 2  6  8 | 3  4  7 | 3  5  8
  1  9 10 | 4  5  9 | 1  8  9 | 2  7  9
  3  8 11 | 3  7 10 | 5  6 11 | 4  6 10
  5  7 12 | 1 11 12 | 2 10 12 | 1 11 12
.
The 8 solutions for m = 4, one per line:
  (1,  5,  6), (2,  8, 10), (3,  9, 12), (4,  7, 11);
  (1,  5,  6), (2,  9, 11), (3,  7, 10), (4,  8, 12);
  (1, 10, 11), (2,  5,  7), (3,  6,  9), (4,  8, 12);
  (1,  6,  7), (2, 10, 12), (3,  8, 11), (4,  5,  9);
  (1,  9, 10), (2,  4,  6), (3,  8, 11), (5,  7, 12);
  (1, 11, 12), (2,  6,  8), (3,  7, 10), (4,  5,  9);
  (1,  8,  9), (2, 10, 12), (3,  4,  7), (5,  6, 11);
  (1, 11, 12), (2,  7,  9), (3,  5,  8), (4,  6, 10).

Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), #10.6.2.
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1,2,...,3n} into Subsets {x,y,z} Satisfying x+y=z, arXiv:2307.00303 [math.CO], 2023.
Matroids Matheplanet, Calculating sequence element a(16) of OEIS A108235
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]
Wikipedia, Dancing Links

Cf. A002848, A002849, A161826, A202951, A202952.

Table[Length[Select[Subsets[Select[Subsets[Range[3 n], {3}], #[[1]] + #[[2]] == #[[3]] &], {n}], Range[3 n] == Sort[Flatten[#]] &]], {n, 0,
5}]  (* Suitable only for n<6. See Knuth's Dancing Links algorithm for n>5. *) (* Robert Price, Apr 03 2019 *)

A = lambda n:sum(1 for t in DLXCPP([(a-1,b-1,a+b-1) for a in (1..3*n) for b in (1..min(3*n-a,a-1))])) # Tomas Boothby, Oct 11 2013

a(n) = 0 unless n == 0 or 1 (mod 4). For n == 0 or 1 (mod 4), a(n) = A002849(3n). See A002849 for references and further information.

a(12) from R. H. Hardin, Feb 11 2010
a(12) confirmed and a(13) computed (using Knuth's dancing links algorithm) by Alois P. Heinz, Feb 11 2010
a(13) confirmed by Tomas Boothby, Oct 11 2013
a(16) from Frank Niedermeyer, Apr 19 2020
a(17)-a(19) from Frank Niedermeyer, May 02 2020

A202705 Number of irreducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms.

Original entry on oeis.org

1, 1, 1, 2, 6, 25, 115, 649, 4046, 29674, 228030, 1987700, 18402704, 188255116, 2030067605, 23829298479, 293949166112, 3909410101509, 54360507919179, 806312701922676

Offset: 0

N. J. A. Sloane, Dec 26 2011

"Irreducible" means that there is no j such that the first j of the triples are a partition of 1, ..., 3j.

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "K".
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] Gives a(0)-a(10).

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

See also A002848, A002849.

G.f.: 2 - 1/g where g is g.f. for A104429. [corrected by Martin Fuller, Jul 08 2025]

a(n) = A279197(n) + 2*A279198(n) for n>0.

a(11)-a(14) from Alois P. Heinz, Dec 28 2011
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
a(18)-a(19) from Martin Fuller, Jul 08 2025

A279197 Number of self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

1, 1, 2, 2, 11, 11, 55, 58, 486, 442, 4218, 3924, 45096, 42013, 538537, 505830, 7368091, 6959545, 111877294, 105723374, 1886636688, 1763443165, 34585786729, 32401780965, 687085545694, 642233156868, 14691047314846, 13788837896728, 340221989868538, 317342350394678, 8327884506579315

Offset: 1

N. J. A. Sloane, Dec 15 2016

nonn

In Richard Guy's letter, the term 50 is marked with a question mark. Peter Kagey has shown that the value should be 55. - N. J. A. Sloane, Feb 15 2017
From Peter Kagey, Feb 14 2017: (Start)
An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (See A202705.)
A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.
(End)

			Examples of solutions X,Y,Z for n=5:
  2,4,3
  5,7,6
  1,15,8
  9,11,10
  12,14,13
and in his letter Richard Guy has drawn links pairing the first and fifth solutions, and the second and fourth solutions.
For n = 2 the a(2) = 1 solution is
  [(2,6,4),(1,5,3)].
For n = 3 the a(3) = 2 solutions are
  [(1,7,4),(3,9,6),(2,8,5)] and
  [(2,4,3),(6,8,7),(1,9,5)].

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "I".
Peter Kagey, Haskell program for A279197.
Peter Kagey, Solutions for a(1)-a(10).
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

See also A002848, A002849.

a(n) = A282616(n) - A282615(n). - Martin Fuller, Jul 15 2025

a(7) corrected and a(8)-a(13) added by Peter Kagey, Feb 14 2017
a(14)-a(16) from Fausto A. C. Cariboni, Feb 27 2017
a(17) from Fausto A. C. Cariboni, Mar 22 2017
a(18)-a(24) from Bert Dobbelaere, May 29 2025
a(25)-a(31) from Martin Fuller, Jul 15 2025

A279199 Number of reducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms: a(n) = A104429(n) - A202705(n).

Original entry on oeis.org

0, 0, 1, 3, 9, 30, 117, 512, 2597, 14892, 99034, 721350, 5909324, 52578654, 516148082, 5422071091, 61889692290, 749456672155, 9767058240577, 134007989313530, 1958535749524107

Offset: 0

N. J. A. Sloane, Dec 15 2016

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "L".
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

See also A002848, A002849.

a(n) = A104429(n)-A202705(n) = Sum_{i=1..n-1} A104429(i)*A202705(n-i). - Martin Fuller, Jul 08 2025

Definition corrected by N. J. A. Sloane, Jan 09 2017 at the suggestion of Fausto A. C. Cariboni.
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
a(18)-a(20) from Martin Fuller, Jul 08 2025

A279198 Number of pairs of conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

0, 0, 0, 2, 7, 52, 297, 1994, 14594, 113794, 991741, 9199390, 94105010, 1015012796, 11914379971, 146974330141, 1954701366709

Offset: 1

N. J. A. Sloane, Dec 15 2016

			Richard Guy gives examples in his letter.

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
Nowakowski, Richard Joseph, Generalization of the Langford-Skolem problem, MS Thesis, University of Calgary, 1975.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "J".
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

See also A002848, A002849.

A279197(n) + 2*A279198(n) = A202705(n).

a(7)-a(16) from Fausto A. C. Cariboni, Feb 27 2017

a(17) from Fausto A. C. Cariboni, Mar 22 2017

A161826 Number of maximal vertex-independent sets in the hypergraph with nodes V = {1, 2, ..., n} and "edges" consisting of the triples (X,Y,Z) with X

Original entry on oeis.org

1, 1, 3, 2, 6, 1, 6, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4

Offset: 1

N. J. A. Sloane, Feb 10 2010

A subset S of V is vertex-independent if there is no edge (X,Y,Z) with X, Y, Z all in S.

Continued fraction expansion of (3452449 + 2*sqrt(2))/1943849. - Stefano Spezia, Mar 17 2024

J. Sedláček, On a set system, Annals New York Acad. Sci., 175 (No. 1, 1970), 329-330.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A002848, A002849, A108235.

a(2k)=1, a(2k+1)=4 for k >= 5.

G.f.: x*(1 + x + 2*x^2 + x^3 + 3*x^4 - x^5 - x^8 - x^10)/((1 - x)*(1 + x)). - Stefano Spezia, Mar 17 2024

cp's OEIS Frontend

A152255 Erroneous version of A002848.

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A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms.

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Examples

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Crossrefs

Extensions

A002849 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n}, each satisfying X + Y = Z.

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Programs

PARI

Extensions

A108235 Number of partitions of {1,2,...,3n} into n triples (X,Y,Z) each satisfying X+Y=Z.

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Programs

Mathematica

Sage

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A202705 Number of irreducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms.

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Formula

Extensions

A279197 Number of self-conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

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Comments

Examples

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Links

Crossrefs

Formula

Extensions

A279199 Number of reducible ways to split 1, 2, 3, ..., 3n into n arithmetic progressions each with 3 terms: a(n) = A104429(n) - A202705(n).

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Author

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Links

Crossrefs

Formula

Extensions

A279198 Number of pairs of conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

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Author

Keywords

Examples

References

Links

Crossrefs

Formula

Extensions

A161826 Number of maximal vertex-independent sets in the hypergraph with nodes V = {1, 2, ..., n} and "edges" consisting of the triples (X,Y,Z) with X