A036239 -id:A036239 - cp's OEIS frontend

A051366 Number of 6-element families of an n-element set such that every 4 members of the family have a nonempty intersection.

0, 0, 0, 0, 112, 39761, 5318420, 506289623, 41378309308, 3133123494417, 227657567966500, 16152548751321851, 1129224692910819164, 78169242144478858373, 5373159786842137703140, 367368738925063893430959

Offset: 0

Vladeta Jovovic, Goran Kilibarda

nonn

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

G. C. Greubel, Table of n, a(n) for n = 0..550

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

Table[1/6! (64^n - 15*60^n + 60*58^n + 25*57^n - 240*56^n + 45*55^n + 705*54^n - 987*53^n - 351*52^n + 3040*51^n - 5445*50^n + 6105*49^n - 4939*48^n + 2997*47^n - 1365*46^n + 455*45^n - 105*44^n + 15*43^n - 42^n - 15*32^n + 75*30^n - 150*29^n + 150*28^n - 75*27^n + 15*26^n + 85*16^n - 85*15^n - 225*8^n + 225*7^n + 274*4^n - 274*3^n - 120*2^n + 120), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

a(n) = (1/6!)*(64^n - 15*60^n + 60*58^n + 25*57^n - 240*56^n + 45*55^n + 705*54^n - 987*53^n - 351*52^n + 3040*51^n - 5445*50^n + 6105*49^n - 4939*48^n + 2997*47^n - 1365*46^n + 455*45^n - 105*44^n + 15*43^n - 42^n - 15*32^n + 75*30^n - 150*29^n + 150*28^n - 75*27^n + 15*26^n + 85*16^n - 85*15^n - 225*8^n + 225*7^n + 274*4^n - 274*3^n - 120*2^n + 120).

A051367 Number of 5-element families of an n-element set such that every 4 members of the family have a nonempty intersection.

Original entry on oeis.org

0, 0, 0, 0, 224, 21281, 1144027, 49310674, 1915317642, 70460566827, 2513684751809, 88008877380908, 3043421159408080, 104321464544910613, 3552122530256316471, 120307381384305672102

Offset: 0

Vladeta Jovovic, Goran Kilibarda

nonn

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

G. C. Greubel, Table of n, a(n) for n = 0..660

Cf. A036239, A051180, A051181, A051182, A051183, A051184, A051185.

[(32^n - 5*30^n + 10*29^n - 10*28^n + 5*27^n - 26^n - 10*16^n + 10*15^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24)/120: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(1/5!)*(32^n - 5*30^n + 10*29^n - 10*28^n + 5*27^n - 26^n - 10*16^n + 10*15^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24), {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((1/5!)*(32^n - 5*30^n + 10*29^n - 10*28^n + 5*27^n - 26^n - 10*16^n + 10*15^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24), ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = (1/5!)*(32^n - 5*30^n + 10*29^n - 10*28^n + 5*27^n - 26^n - 10*16^n + 10*15^n + 35*8^n - 35*7^n - 50*4^n + 50*3^n + 24*2^n - 24).

A051375 Number of Boolean functions of n variables and rank 3 from Post class F(5,inf).

Original entry on oeis.org

0, 0, 9, 66, 345, 1590, 6909, 29106, 120465, 493230, 2005509, 8116746, 32744985, 131801670, 529647309, 2125861986, 8525167905, 34165634910, 136857036309, 548010848826, 2193789933225, 8780396200950, 35137287916509

Offset: 1

Vladeta Jovovic, Goran Kilibarda

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A036240.

[(4^n - 3^n - 3*2^n + 5)/2: n in [0..50]]; // G. C. Greubel, Oct 08 2017

Table[(4^n - 3^n - 3*2^n + 5)/2, {n, 0, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=0,50, print1((4^n - 3^n - 3*2^n + 5)/2, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = A036239(n) - A000918(n).
a(n) = (4^n - 3^n - 3*2^n + 5)/2.
a(n) = Sum_{j=1..n} (-1)^(j+1)*C(n, j)*C(2^(n-j)-1, k-1) (with k=3).
Also: 1/(k-1)!*Sum(s(k, j)*(2^((j-1)*n)-(2^(j-1)-1)^n), j=1..k), where s(k, j) are Stirling numbers of the first kind (with k=3).
From Colin Barker, Jun 25 2012: (Start)
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4).
G.f.: 3*x^3*(3-8*x)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)). (End)

More terms from James Sellers

A051376 Number of Boolean functions of n variables and rank 4 from Post class F(5,inf).

Original entry on oeis.org

0, 0, 3, 134, 1935, 20830, 198303, 1776894, 15402495, 130890110, 1098087903, 9130126654, 75412301055, 619706950590, 5071742430303, 41369422556414, 336511166127615, 2730929153686270, 22119108433729503, 178853777028618174

Offset: 1

Vladeta Jovovic, Goran Kilibarda

G. C. Greubel, Table of n, a(n) for n = 1..1000
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (25,-241,1135,-2734,3160,-1344).

Cf. A000918, A036239, A036240.

[(8^n - 7^n - 6*4^n + 6*3^n + 11*2^n - 17)/6: n in [1..50]]; // G. C. Greubel, Oct 08 2017

Table[(8^n - 7^n - 6*4^n + 6*3^n + 11*2^n - 17)/6, {n, 1, 50}] (* G. C. Greubel, Oct 08 2017 *)

for(n=1,50, print1((8^n - 7^n - 6*4^n + 6*3^n + 11*2^n - 17)/6, ", ")) \\ G. C. Greubel, Oct 08 2017

a(n) = A036240(n) - A036239(n) + A000918(n).
a(n) = (8^n - 7^n - 6*4^n + 6*3^n + 11*2^n - 17)/6.
a(n) = Sum_{j=1..n} (-1)^(j+1)*C(n, j)*C(2^(n-j)-1, k-1), where k=4.
Also: 1/(k-1)!*Sum_{j=1..k} s(k, j)*(2^((j-1)*n)-(2^(j-1)-1)^n), where s(k, j) are Stirling numbers of the first kind (and k=4).
G.f.: x^3*(3 + 59*x - 692*x^2 + 1344*x^3) / ( (x-1)*(4*x-1)*(3*x-1)*(2*x-1)*(8*x-1)*(7*x-1) ). - R. J. Mathar, Jun 13 2013

More terms from James Sellers

A051381 Number of Boolean functions of n variables from Post class F(5,inf).

Original entry on oeis.org

1, 3, 19, 471, 162631, 12884412819, 64563604212887416603, 1361129467683753853595244012815395920687, 521064401567922879406069432539095585333589848390805645835993148352662477920015

Offset: 1

Vladeta Jovovic, Goran Kilibarda

G. C. Greubel, Table of n, a(n) for n = 1..12
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.
S. Spasovski and A. M. Bogdanova, Optimization of the Polynomial Greedy Solution for the Set Covering Problem, 2013, 10th Conference for Informatics and Information Technology (CIIT 2013).
Index entries for sequences related to Boolean functions

Cf. A036239, A036240. Equals A005530(n)/2.

Table[Sum[(-1)^(j + 1)*Binomial[n, j]*2^(2^(n - j) - 1) , {j, 1, n}], {n, 1, 5}] (* G. C. Greubel, Oct 08 2017 *)

a(n) = Sum_{j=1..n} (-1)^(j+1)*C(n, j)*2^(2^(n-j)-1).

More terms from James Sellers

A059090 Triangle T(n,m) giving number of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,.., A037952(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 3, 1, 1, 15, 30, 30, 5, 1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1, 1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Dec 28 2000

An antichain is called intersecting (or proper) antichain if every two members have a nonempty intersection. Row sums give the number of intersecting antichains on a labeled n-set or n-variable Boolean functions in the Post class F(7,2) or self-dual monotone Boolean functions of n+1 variables. Cf. A001206.

			1;
1, 1;
1, 3;
1, 7, 3, 1;
1, 15, 30, 30, 5;
1, 31, 195, 605, 780, 543, 300, 135, 45, 10, 1;
1, 63, 1050, 9030, 41545, 118629, 233821, 329205, 327915, 224280, 100716, 29337, 5950, 910, 105, 7;

Jovovic V., Kilibarda G., The number of n-variable Boolean functions in the Post class F(7,2), Belgrade, 2001, in preparation.
Pogosyan G., Miyakawa M., Nozaki A., Rosenberg I., The Number of Clique Boolean Functions, IEICE Trans. Fundamentals, Vol. E80-A, No. 8, pp. 1502-1507, 1997/8.

Cf. A001206, A032263, A051303-A051307, A036239, A051180-A051185, A016269, A047707, A051112-A051118, A000372.

T(n, 0)=1, T(n, 1)=2^n-1, T(n, 2)=A032263(n), T(n, 3)=A051303(n), T(n, 4)=A051304(n), T(n, 5)=A051305(n), T(n, 6)=A051306(n), T(n, 7)=A051307(n).

A374453 Number of set-systems S composed of nonempty subsets of [n] such that any element k in S appears at most k times.

Original entry on oeis.org

1, 2, 6, 36, 547, 26672, 5120069, 4581266029, 21912279450653, 627026135401140277, 118043015040470215561725, 158758107128989643461422723149, 1641097327889006717487651007699748392

Offset: 0

John Tyler Rascoe, Jul 08 2024

			In the set-system: {{2, 3, 5},{3, 4, 5},{1, 3},{2}} no element k appears more than k times, so it is counted under a(5) = 26672.
a(n) for n = 0..2 counts the following set-systems:
a(0) = 1: {}.
a(1) = 2: {}, {{1}}.
a(2) = 6: {}, {{1}}, {{2}}, {{1},{2}}, {{1,2}}, {{1,2},{2}}.

John Tyler Rascoe, Python program.

Cf. A036239, A072444, A072445, A306021, A323818.

Python
```
# see linked program
```

a(7)-a(8) from Alois P. Heinz, Jul 09 2024
a(9) from Alois P. Heinz, Jul 15 2024
a(10)-a(12) from Christian Sievers, Jul 17 2024

A051368 Number of Boolean functions of n variables and rank 8 from the Post class F(5,2).

Original entry on oeis.org

0, 0, 0, 12, 105765, 59046810, 16636450912, 3491313542424, 627725748292995, 102894277877828670, 15867914519581210614, 2343602605748557069356, 335205287948366997151705, 46782266953279485879549090

Offset: 1

Vladeta Jovovic, Goran Kilibarda

nonn

E. Post, Two-valued iterative systems, Annals of Mathematics, no 5, Princeton University Press, NY, 1941.
V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Index entries for sequences related to Boolean functions

A051184(n)-A051183(n)+A051182(n)-A051181(n)+A051180(n)-A036239(n)+A000918(n)

A308700 a(n) = n * 2^(n - 2) * (2^(n - 1) - 1).

Original entry on oeis.org

0, 0, 2, 18, 112, 600, 2976, 14112, 65024, 293760, 1308160, 5761536, 25153536, 109025280, 469704704, 2013143040, 8589672448, 36506664960, 154617643008, 652832538624, 2748773826560, 11544861081600, 48378488553472, 202310091276288, 844424829468672, 3518436999168000

Offset: 0

Stefano Spezia, Jun 17 2019

Given a pseudo-graph P of the set X = {1, 2, ..., n}, defined as a graph represented by the discrete topology on the set X (the power set of X), for n > 0, a(n) is the number of edges of the topological graph arising by deleting loops in P (see Theorem 3.3 in Kozae et al.).

			For n = 3, the set X = {1,2,3},
  the power set 2^X = {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, X} and the pseudo-graph P represented by 2^X has the following edges, here grouped into...
  simple loops:
  {1} --- {1}, {2} --- {2}, {3} --- {3} for a total of 3.
  double loops:
  {1,2} --- {1,2}, {1,3} --- {1,3}, {2,3} --- {2,3} for a total of 6 simple loops.
  triple loop:
  X --- X for a total of 3 simple loops.
  simple edges:
  {1} --- {1,2}, {1} --- {1,3}, {1} --- X, {2} --- {1,2}, {2} --- {2,3}, {2} --- X, {3} --- {1,3}, {3} --- {2,3}, {3} --- X, {1,2} --- {1,3}, {1,2} --- {2,3}, {1,3} --- {2,3} for a total of 12.
  double edges:
  {1,2} --- X, {1,3} --- X, {2,3} --- X for a total of 6 simple edges.
  By deleting the loops in P, there remain a total of a(3) = 12 + 6 = 18 edges for the topological graph arising from P.

A. M. Kozae, A. A. El Atik, A. Elrokh and M. Atef, New types of graphs induced by topological spaces, Journal of Intelligent & Fuzzy Systems, vol. 36, no. 6 (2019), pp. 5125-5134; on Research Gate.
Index entries for linear recurrences with constant coefficients, signature (12,-52,96,-64).

Cf. A000225, A001787, A005843, A006516, A036239, A082134, A171476, A171496.

Cf. A082134 (total number of edges of the pseudo-graph P).

Flat(List([0..25], n->n*2^(n-2)*(2^(n-1)-1)))

Magma
```
[n*2^(n-2)*(2^(n-1)-1): n in [0..25]];
```

a:=n->n*2^(n-2)*(2^(n-1)-1): seq(a(n),n=0..25);

Table[n 2^(n - 2)(2^(n - 1) - 1), {n, 0, 31}]

makelist(n*2^(n-2)*(2^(n-1)-1), n, 0, 25);

PARI
```
a(n)=n*2^(n-2)*(2^(n-1)-1);
```

O.g.f.: -2 * x^2 * (-1 + 3*x)/((-1 + 2*x)^2 * (-1 + 4*x)^2).
E.g.f.: (1/2) * exp(2*x) * (-1 + exp(2*x)) * x.
a(n) = 12 * a(n - 1) - 52*a(n - 2) + 96*a(n - 3) - 64*a(n - 4) for n > 3.
a(n) = n * 2^(n - 2) * (2^(n - 1) - 1).
Lim_{n -> infinity} a(n)/a(n - 1) = 4.
a(n) = A082134(n) - A001787(n).
a(n) = A005843(A001787(n)) * A000225(n - 1).
a(n) = n * A006516(n - 1).
a(n) = n * A171476(n - 2).
a(n) = n * A171496(n - 3).

A344494 Triangle read by rows: The d-th row contains the Betti numbers of the d-dimensional resonance arrangement.

Original entry on oeis.org

1, 3, 2, 7, 15, 9, 15, 80, 170, 104, 31, 375, 2130, 5270, 3485, 63, 1652, 22435, 159460, 510524, 371909, 127, 7035, 215439, 3831835, 37769977, 169824305, 135677633, 255, 29360, 1957200, 81029004, 2076831708, 30623870732, 207507589302, 178881449368, 511, 120975, 17153460, 1582492380, 96834110730, 3829831100340, 89702833260450, 973784079284874, 887815808473419

Offset: 1

Lukas Kühne, May 21 2021

a(d,i) is the i-th Betti number of the d-dimensional resonance arrangement (for 1 <= i <= d).
The d-dimensional resonance arrangement is the hyperplane arrangement in the d-dimensional space (x_1,...,x_d) consisting of (2^d - 1) hyperplanes c_1*x_1 + c_2*x_2 + ... + c_d*x_d = 0 where c_j are 0 or +1 and we exclude the case with all c=0. This arrangement is also called the all-subset arrangement.
The Betti numbers are also called Whitney numbers of the second kind and they are also the absolute values of the coefficients of the characteristic polynomial of the arrangement.
The sum of the Betti numbers equals the number of chambers of this arrangement.
The Betti numbers for the 8- and 9-dimensional resonance arrangement were computed with the julia package CountingChambers.jl.

			Triangle begins
   1;
   3,   2;
   7,  15,    9;
  15,  80,  170,  104;
  31, 375, 2130, 5270, 3485;

T. Brysiewicz, H. Eble, and L. Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021 (provides the Betti numbers for the d-dimensional resonance arrangement for 1<=d<=9).
Z. Chroman and M. Singhal, Computations associated with the resonance arrangemnt, arXiv:2106.09940 [math.CO], 2021 (provides a formula for the fourth Betti number of this arrangement and the Betti numbers for the d-dimensional resonance arrangement for 1<=d<=9).
H. Kamiya, A. Takemura, and H. Terao, Ranking patterns of unfolding models of codimension one, Advances in Applied Mathematics 47 (2011) 379 - 400 (provides the Betti numbers for the d-dimensional resonance arrangement for 1<=d<=7).
Lukas Kühne, The Universality of the Resonance Arrangement and its Betti Numbers, arXiv:2008.10553 [math.CO], 2020 (provides formula for the second and third Betti number of this arrangement).

A034997 is the sum of each row (Number of generalized retarded functions in quantum field theory).
A000225 is the first column (2^d - 1).
A036239 is the second column (1/2) * (4^n - 3^n - 2^n + 1).

cp's OEIS Frontend

A051366 Number of 6-element families of an n-element set such that every 4 members of the family have a nonempty intersection.

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A051367 Number of 5-element families of an n-element set such that every 4 members of the family have a nonempty intersection.

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A051375 Number of Boolean functions of n variables and rank 3 from Post class F(5,inf).

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A051376 Number of Boolean functions of n variables and rank 4 from Post class F(5,inf).

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A051381 Number of Boolean functions of n variables from Post class F(5,inf).

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A059090 Triangle T(n,m) giving number of m-element intersecting antichains on a labeled n-set or n-variable Boolean functions with m nonzero values in the Post class F(7,2), m=0,.., A037952(n).

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A374453 Number of set-systems S composed of nonempty subsets of [n] such that any element k in S appears at most k times.

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Python

Extensions

A051368 Number of Boolean functions of n variables and rank 8 from the Post class F(5,2).

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