A028243 -id:A028243 - cp's OEIS frontend

A156235 Number of irreflexive binary relations on the power set P(N) of an n-element set N as restricted below.

1, 1, 4, 198, 209342

Offset: 0

Rick L. Shepherd, Feb 06 2009

Each enumerated irreflexive relation R has these restricting properties:
Let (A,B) and (C,D) be arbitrary elements of R. Then
i) A and B are nonempty subsets of N,
ii) A and B are disjoint, and
iii) if (A,B) is not equal to (C,D) and A intersect C is nonempty, then B and D are disjoint.
Each a(n) includes the empty relation. Each relation R may contain any number of elements from 0 to n^2-n.
Inspired by considering less-restricted gift-exchange scenarios than in A053763.
Essentially, the scenarios here relax (somewhat but not entirely) noted restrictions iii) and iv) given there to allow joint giving and joint receiving.
More generally, these relations could be considered distribution networks (or even possible economies, in some sense) for goods and/or services whenever an entity cannot directly distribute to itself or to another entity of which it is a part and whenever an entity cannot (jointly) distribute directly to a second entity in more than one way (e.g., as part of two larger entities).

			One of the 209342 irreflexive relations corresponding to a(4) is
R = {({1},{2}), ({2},{1}), ({3,4},{1,2}), ({1,4},{3}), ({2},{3,4})}.
Notice how the last three ordered pairs correspond to jointly giving and/or receiving gifts.

Cf. A000392, A002378, A028243, A053763.

A245020 Number of ordered n-tuples of positive integers, whose minimum is 0 and maximum is 5.

Original entry on oeis.org

0, 2, 30, 302, 2550, 19502, 140070, 963902, 6433590, 41983502, 269335110, 1705278302, 10686396630, 66425568302, 410223570150, 2520229093502, 15417960407670, 93999281613902, 571487645261190, 3466523088409502, 20987674370482710, 126870924446280302

Offset: 1

Ovidiu Bagdasar, Sep 17 2014

For given k and n positive integers, let T(k,n) represent the number of n-tuples of positive integers, whose minimum is zero and maximum is k. In this notation, the sequence corresponds to a(n) = T(5,n).

			For n=2 the a(2)=2 solutions are (0,5) and (5,0).

O. Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples, Scientific publications of the state university of Novi Pazar, Ser. A: Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91--100.
Index entries for linear recurrences with constant coefficients, signature (15,-74,120).

T(1,n) gives A000918; T(2,n-1) gives A028243, T(n,3) gives A008588, T(n,4) gives A005914.

Cf. A016103.

LinearRecurrence[{15,-74,120},{0,2,30},30] (* Harvey P. Dale, Nov 20 2020 *)

concat(0, Vec(-2*x^2/((4*x-1)*(5*x-1)*(6*x-1)) + O(x^100))) \\ Colin Barker, Sep 18 2014

a(n) = 6^n-2*5^n+4^n.
a(n) = 15*a(n-1)-74*a(n-2)+120*a(n-3) for n>3. G.f.: -2*x^2 / ((4*x-1)*(5*x-1)*(6*x-1)). - Colin Barker, Sep 18 2014
a(n) = 2*A016103(n). - Colin Barker, Sep 18 2014

A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 7, 10, 14, 19, 15, 22, 32, 46, 65, 31, 46, 68, 100, 146, 211, 63, 94, 140, 208, 308, 454, 665, 127, 190, 284, 424, 632, 940, 1394, 2059, 255, 382, 572, 856, 1280, 1912, 2852, 4246, 6305, 511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171

Offset: 0

Paul Curtz, Jun 20 2025

			Triangle begins:
    0;
    1,   1;
    3,   4,    5;
    7,  10,   14,   19;
   15,  22,   32,   46,   65;
   31,  46,   68,  100,  146,  211;
   63,  94,  140,  208,  308,  454,  665;
  127, 190,  284,  424,  632,  940, 1394, 2059;
  255, 382,  572,  856, 1280, 1912, 2852, 4246,  6305;
  511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171;
  ...

Columns k=0..2: A000225, A033484, A053209 (sans 1).
Diagonals: A001047, A027649, A053581 (sans 1), A291012 (sans 2).
Cf. A028243 (row sums), A036561, A059268, A081656, A248216, A321003.

/* As triangle */ [[2^(n-k)*3^k - 2^k : k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jun 27 2025

T:= proc(n,k) option remember;
     `if`(n=k, 3^n-2^n, T(n, k+1)-T(n-1, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2025

t[n_, 0] := 3^n - 2^n; t[n_, k_] := t[n, k] = t[n + 1, k - 1] - t[n, k - 1]; Table[t[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 20 2025 *)

T(n,n) = 3^n - 2^n = A001047(n).
T(n,k) = T(n,k+1) - T(n-1,k) for 0 <= k < n.
T(n,k) = 2^(n-k)*3^k - 2^k = A036561(n,k) - A059268(n,k).
T(2n,n) = A248216(n+1).

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A156235 Number of irreflexive binary relations on the power set P(N) of an n-element set N as restricted below.

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A245020 Number of ordered n-tuples of positive integers, whose minimum is 0 and maximum is 5.

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A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

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