A000311 -id:A000311 - cp's OEIS frontend

A319121 Number of complete multimin tree-factorizations of Heinz numbers of integer partitions of n.

1, 2, 5, 18, 74, 344, 1679, 8548, 44690, 238691, 1295990, 7132509

Offset: 1

Gus Wiseman, Sep 11 2018

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of at least two multimin tree-factorizations, one of each factor in a multimin factorization of n. A multimin tree-factorization is complete if the leaves are all prime numbers.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(3) = 5 trees are: 5, (2*3), (2*2*2), (2*(2*2)), ((2*2)*2).
The a(4) = 18 trees (normalized with prime(n) -> n):
  4,
  (13), (22), (112), (1111),
  (1(12)), ((12)1), ((11)2),
  (11(11)), (1(11)1), ((11)11), (1(111)), ((111)1), ((11)(11)),
  (1(1(11))), (1((11)1)), ((1(11))1), (((11)1)1).

Cf. A000311, A001003, A001055, A020639, A255397, A281113, A281118, A281119, A295281, A317545, A317546, A318577, A319118, A319119.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Sum[Length[Select[mmftrees[k],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{k,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,10}]

a(11)-a(12) from Robert Price, Sep 14 2018

A330785 Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 8, 4, 0, 1, 9, 25, 28, 11, 0, 1, 13, 57, 111, 99, 33, 0, 1, 20, 129, 379, 561, 408, 116, 0, 1, 28, 253, 1057, 2332, 2805, 1739, 435, 0, 1, 40, 496, 2833, 8695, 15271, 15373, 8253, 1832, 0, 1, 54, 898, 6824, 28071, 67790, 98946, 85870, 40789, 8167

Offset: 1

Gus Wiseman, Jan 03 2020

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   3   2
   0   1   5   8   4
   0   1   9  25  28  11
   0   1  13  57 111  99  33
   0   1  20 129 379 561 408 116
Row n = 5 counts the following chains (minimum and maximum not shown):
  ()  (14)    (113)->(14)    (1112)->(113)->(14)
      (23)    (113)->(23)    (1112)->(113)->(23)
      (113)   (122)->(14)    (1112)->(122)->(14)
      (122)   (122)->(23)    (1112)->(122)->(23)
      (1112)  (1112)->(14)
              (1112)->(23)
              (1112)->(113)
              (1112)->(122)

Row sums are A213427.
Main diagonal is A002846.
Column k=3 is A007042.
Dominated by A330784.
The version for set partitions is A008826.
The version for factorizations is A330935.
Cf. A000111, A000258, A000311, A005121, A141268, A196545, A265947, A300383, A306186, A317141, A317176, A318813, A320160, A330679.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
upr[q_]:=Union[Sort/@Apply[Plus,mps[q],{2}]];
paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y,#}&/@DeleteCases[upr[y],y],{y,Sort/@IntegerPartitions[n]}],ConstantArray[1,n],{n}],Length[#]==k-1&]],{n,8},{k,n}]

T(n,k) = A330935(2^n,k).

A364026 Table read by descending antidiagonals. T(n,k) is the big Ramsey degree of k in w^n, where w is the first transfinite ordinal, omega.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 4, 1, 1, 0, 1, 26, 14, 1, 1, 0, 1, 236, 509, 49, 1, 1, 0, 1, 2752, 35839, 10340, 175, 1, 1, 0, 1, 39208, 4154652, 5941404, 222244, 637, 1, 1, 0, 1, 660032, 718142257, 7244337796, 1081112575, 4981531, 2353, 1, 1, 0, 1, 12818912, 173201493539

Offset: 0

Nathan Hurtig, Jul 01 2023

T(n,k) is the least integer t such that, for all finite colorings of the k-subsets of w^n, there exists some S, an order-equivalent subset to w^n, where that coloring restricted to the k-subsets of S outputs at most t colors.
By Ramsey's theorem, the first row T(1,k)=1 for all k.
The second row T(2,k) coincides with A000311.
The second column T(n,2) coincides with A079309.

			The data is organized in a table beginning with row n = 0 and column k = 0. The data is read by descending antidiagonals. T(2,3)=26.
The table T(n,k) begins:
[n/k]   0   1      2        3       4         5   ...
--------------------------------------------------------------------
[0]     1,  1,     0,       0,      0,        0,  ...
[1]     1,  1,     1,       1,      1,        1,  ...
[2]     1,  1,     4,      26,    236,     2572,  ...
[3]     1,  1,    14,     509,  35839,  4154652,  ...
[4]     1,  1,    49,   10340,  ...
[5]     1,  1,   175,  222244,  ...
[6]     1,  1,   637,  ...

Dragan Mašulovic and Branislav Šobot, Countable ordinals and big Ramsey degrees, Combinatorica, 41 (2021), 425-446.
Alexander S. Kechris, Vladimir G. Pestov, and Stevo Todorčević, Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups, Geometric & Functional Analysis, 15 (2005), 106-189.

Joanna Boyland, William Gasarch, Nathan Hurtig, and Robert Rust, Big Ramsey Degrees of Countable Ordinals, arXiv:2305.07192 [math.CO], 2023.
Eric Weisstein's World of Mathematics, Ordinal Number.

T(2,k) is A000311. T(n,2) is A079309.

pp p n k
  | n == 0 && k >= 2 = 0
  | k == 0 && p == 0 = 1
  | k == 0 && p >= 1 = 0
  | n == 0 && k == 1 && p == 0 = 1
  | n == 0 && k == 1 && p >= 1 = 0
  | n == 1 && k >= 1 && k == p = 1
  | n == 1 && k >= 1 && k /= p = 0
  | n >= 2 && k >= 1 = sum [binom (p-1) i * pp i (n-1) j * pp (p-1-i) n (k-j) | i <- [0..p-1], j <- [1..k]]
binom n 0 = 1
binom 0 k = 0
binom n k = binom (n-1) (k-1) * n `div` k
a364026 n k =
  sum [pp p n k | p <- [0..n*k]]

T(n,k) = Sum_{p=0..n*k} P(p,n,k), where for n >= 2 and k >= 1,
P(0,n,k) = 0, and for p >= 1,
P(p,n,k) = Sum_{j=1..k} Sum_{0..p-1} binomial(p-1,i)*P(i,n-1,j)*P(p-1-i,n,k-j).

A060694 A triangle related to rooted trees.

Original entry on oeis.org

1, 2, 1, 10, 8, 2, 82, 86, 36, 6, 938, 1202, 668, 192, 24, 13778, 20772, 14118, 5452, 1200, 120, 247210, 427828, 341122, 161688, 48312, 8640, 720, 5240338, 10228458, 9325398, 5184902, 1909920, 467784, 70560, 5040, 128149802, 278346286

Offset: 1

F. Chapoton, Apr 20 2001

The rows sum to A006963, the alternating sum is A000311, the right column is A000142, the left column is related to A032188 (twice); the second-to-right column is A052582

			{1}, {2,1}, {10,8,2}, {82,86,36,6}, {938,1202,668,192,24}

Cf. A006963, A000311, A000142, A032188, A052582.

E.g.f. given by the Maple expression RootOf(-exp(_Z*x*t)+x*t*exp(_Z*x*t)+y*t*exp(-_Z+_Z*x*t)-y*t^2*x*exp(-_Z+_Z*x*t)+1-t+t*exp(-_Z+_Z*x*t)-x*t*exp(-_Z+_Z*x*t));

More terms from Vladeta Jovovic, Apr 21 2001

A207326 E.g.f. satisfies 2*A(x)-exp(A(x))+1=sin(x).

Original entry on oeis.org

1, 1, 3, 22, 197, 2248, 31311, 514592, 9750137, 209265504, 5018169035, 132971582976, 3858414981085, 121679533902592, 4143895711622327, 151566138479037952, 5925617619735873969

Offset: 1

Vladimir Kruchinin, Feb 17 2012

nonn

Rest[CoefficientList[Series[(-1 - 2*LambertW[-E^((Sin[x]-1)/2)/2] + Sin[x])/2,{x,0,20}],x] * Range[0,20]!] (* Vaclav Kotesovec, Aug 04 2014 *)

a(n)=sum((m=0..(n-1)/2, v(n-2*m)*sum(i=0..(n-2*m)/2, (2*i+2*m-n)^n*binomial(n-2*m,i)*(-1)^(n+m-i)))/(2^(n-2*m-1)*(n-2*m)!)), v(n)=A000311(n).

a(n) ~ ((1-log(2))*log(2))^(1/4) * n^(n-1) / ((arcsin(2*log(2)-1))^(n-1/2) * exp(n)). - Vaclav Kotesovec, Aug 04 2014

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A319121 Number of complete multimin tree-factorizations of Heinz numbers of integer partitions of n.

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A330785 Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.

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A364026 Table read by descending antidiagonals. T(n,k) is the big Ramsey degree of k in w^n, where w is the first transfinite ordinal, omega.

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A060694 A triangle related to rooted trees.

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A207326 E.g.f. satisfies 2*A(x)-exp(A(x))+1=sin(x).

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