A294201 -id:A294201 - cp's OEIS frontend

A152175 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations.

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 2, 1, 1, 7, 18, 13, 3, 1, 1, 9, 43, 50, 20, 3, 1, 1, 19, 126, 221, 136, 36, 4, 1, 1, 29, 339, 866, 773, 296, 52, 4, 1, 1, 55, 946, 3437, 4281, 2303, 596, 78, 5, 1, 1, 93, 2591, 13250, 22430, 16317, 5817, 1080, 105, 5, 1, 1, 179, 7254, 51075, 115100, 110462, 52376, 13299, 1873, 147, 6, 1

Offset: 1

Vladeta Jovovic, Nov 27 2008

Number of n-bead necklace structures using exactly k different colored beads. Turning over the necklace is not allowed. Permuting the colors does not change the structure. - Andrew Howroyd, Apr 06 2017

			Triangle begins with T(1,1):
  1;
  1,   1;
  1,   1,     1;
  1,   3,     2,      1;
  1,   3,     5,      2,      1;
  1,   7,    18,     13,      3,      1;
  1,   9,    43,     50,     20,      3,      1;
  1,  19,   126,    221,    136,     36,      4,      1;
  1,  29,   339,    866,    773,    296,     52,      4,     1;
  1,  55,   946,   3437,   4281,   2303,    596,     78,     5,    1;
  1,  93,  2591,  13250,  22430,  16317,   5817,   1080,   105   , 5,   1;
  1, 179,  7254,  51075, 115100, 110462,  52376,  13299,  1873,  147,   6, 1;
  1, 315, 20125, 194810, 577577, 717024, 439648, 146124, 27654, 3025, 187, 6, 1;
  ...
For T(4,2)=3, the set partitions are AAAB, AABB, and ABAB.
For T(4,3)=2, the set partitions are AABC and ABAC.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Tilman Piesk, Partition related number triangles

Columns 2-6 are A056295, A056296, A056297, A056298, A056299.
Row sums are A084423.
Partial row sums include A000013, A002076, A056292, A056293, A056294.
Cf. A075195, A087854, A008277 (set partitions), A284949 (up to reflection), A152176 (up to rotation and reflection).
A(1,n,k) in formula is the Stirling subset number A008277.
A(2,n,k) in formula is A293181; A(3,n,k) in formula is A294201.

(* Using recursion formula from Gilbert and Riordan:*)
Adn[d_, n_] := Adn[d, n] = Which[0==n, 1, 1==n, DivisorSum[d, x^# &],
  1==d, Sum[StirlingS2[n, k] x^k, {k, 0, n}],
  True, Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n - 1], x] x]];
Table[CoefficientList[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/(x n), x],
   {n, 1, 10}] // Flatten (* Robert A. Russell, Feb 23 2018 *)
Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d,Adnk[d,n-1,k-#] &], Boole[n==0 && k==0]]
Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/n,{n,1,12},{k,1,n}] // Flatten (* Robert A. Russell, Oct 16 2018 *)

\\ see A056391 for Polya enumeration functions
T(n,k) = NonequivalentStructsExactly(CyclicPerms(n), k); \\ Andrew Howroyd, Oct 14 2017

R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
{ my(A=R(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019

T(n,k) = (1/n)*Sum_{d|n} phi(d)*A(d,n/d,k), where A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)). - Robert A. Russell, Oct 16 2018

A002874 The number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles.

Original entry on oeis.org

1, 2, 8, 42, 268, 1994, 16852, 158778, 1644732, 18532810, 225256740, 2933174842, 40687193548, 598352302474, 9290859275060, 151779798262202, 2600663778494172, 46609915810749130, 871645673599372868, 16971639450858467002, 343382806080459389676

Offset: 0

N. J. A. Sloane, Simon Plouffe

Original name: Sorting numbers.

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).

Alois P. Heinz, Table of n, a(n) for n = 0..484 (first 101 terms from T. D. Noe)
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
J. Pasukonis, S. Ramgoolam, From counting to construction for BPS states in N=4 SYM, arXiv:1010.1683 [hep-th], 2010, (E.3).
J. Pasukonis, S. Ramgoolam, From counting to construction for BPS states in N=4 SYM, J. High En. Phys. 2011 (2) (2011), (E.3).
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

u[n,j] generates for j=1, A000110; j=2, A002872; j=3, this sequence; j=4, A141003; j=5, A036075; j=6, A141004; j=7, A036077. - Wouter Meeussen, Dec 06 2008
Equals column 3 of A162663. - Michel Marcus, Mar 27 2013
Row sums of A294201.

S:= series(exp( (exp(3*x) - 4)/3 + exp(x)), x, 31):
seq(coeff(S,x,j)*j!, j=0..30); # Robert Israel, Oct 30 2015
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((1+
      3^(j-1))*binomial(n-1, j-1)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 17 2017

u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; Table[u[n,3],{n,0,12}] (* Wouter Meeussen, Dec 06 2008 *)
mx = 16; p = 3; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 3^k * BellB[k, 1/3] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: exp( (exp(3*x) - 4)/3 + exp(x) ).
a(n) ~ exp(exp(3*r)/3 + exp(r) - 4/3 - n) * (n/r)^(n + 1/2) / sqrt((1 + 3*r)*exp(3*r) + (1 + r)*exp(r)), where r = LambertW(3*n)/3 - 1/(1 + 3/LambertW(3*n) + n^(2/3) * (1 + LambertW(3*n)) * (3/LambertW(3*n))^(5/3)). - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (3*n/LambertW(3*n))^n * exp(n/LambertW(3*n) + (3*n/LambertW(3*n))^(1/3) - n - 4/3) / sqrt(1 + LambertW(3*n)). - Vaclav Kotesovec, Jul 10 2022

New name from Danny Rorabaugh, Oct 24 2015

A002875 Sorting numbers (see Motzkin article for details).

Original entry on oeis.org

1, 2, 4, 24, 128, 880, 7440

Offset: 0

N. J. A. Sloane

How is the sequence defined (see the links in A000262)? Also more terms would be welcome.

Based on the Motzkin article, where this sequence appears in the last row of the table on p. 173, one would expect that this sequence is the same as A294202. However, they seem to be unrelated. So the true definition of this sequence is a mystery. - Andrew Howroyd and Andrey Zabolotskiy, Oct 25 2017

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).

Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A000262, A001861, A002872, A002873, A002874, A036073, A294201, A294202.

A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

Original entry on oeis.org

1, 3, 10, 33, 106, 333, 1030, 3153, 9586, 29013, 87550, 263673, 793066, 2383293, 7158070, 21490593, 64504546, 193579173, 580868590, 1742867913, 5229128026, 15688432653, 47067395110, 141206379633, 423627527506, 1270899359733

Offset: 1

Vladeta Jovovic and Goran Kilibarda, Feb 28 2000

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 3) x = y. - Ross La Haye, Jan 12 2008
From Paul Barry, Apr 27 2003: (Start)
With offset 0, this is a(n) = (3*3^n - 2*2^n + 1)/2.
G.f. (1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f. (3*exp(3*x) - 2*exp(2*x) + exp(x))/2.
Binomial transform of A083329.
Second binomial transform of A040001. (End)

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, in Russian, 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, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000225, A000392, A028243, A000079.
Cf. A036239.
Column k=2 of A288638.
Third column of A294201.

[(3^n-2^n+1)/2: n in [1..30]]; // G. C. Greubel, Oct 06 2017

A053156:=n->(3^n - 2^n + 1)/2: seq(A053156(n), n=1..40); # Wesley Ivan Hurt, Oct 06 2017

LinearRecurrence[{6,-11,6}, {1, 3, 10}, 50] (* or *) Table[(3^n - 2^n + 1)/2, {n,1,50}] (* G. C. Greubel, Oct 06 2017 *)

a(n) = (3^n-2^n+1)/2; \\ Michel Marcus, Nov 30 2015

a(n) = (3^n - 2^n + 1)/2.
a(n) = StirlingS2(n+2,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 12 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
G.f.: x*(1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)). (End)

A294202 The maximal number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles, and which have the same number of nonempty parts.

Original entry on oeis.org

1, 1, 3, 12, 67, 465, 3675, 30024, 299250, 3417690, 38983966, 446295630, 6494597538, 95113861987, 1365645758568, 20909896016688, 373941213111567, 6583031224561656, 114432377809889706, 2158725804226303597, 45003872172663258463, 928103099363098553160

Offset: 0

Andrew Howroyd, Oct 24 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Maximum row values of A294201.

Cf. A002873.

G(n)={Vec(serlaplace(exp(sumdiv(3, d, y^d*(exp(d*x + O(x*x^n))-1)/d))))}
seq(n)={my(A=G(n)); vector(#A, n, vecmax(Vec(A[n])))} \\ Andrew Howroyd, Sep 20 2019

a(0)=1 prepended by Andrew Howroyd, Sep 20 2019

cp's OEIS Frontend

A152175 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations.

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A002874 The number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles.

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A002875 Sorting numbers (see Motzkin article for details).

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A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

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Magma

Maple

Mathematica

PARI

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A294202 The maximal number of partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles, and which have the same number of nonempty parts.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions