A273873 -id:A273873 - cp's OEIS frontend

A289501 Number of enriched p-trees of weight n.

1, 1, 2, 4, 12, 32, 112, 352, 1296, 4448, 16640, 59968, 231168, 856960, 3334400, 12679424, 49991424, 192890880, 767229952, 2998427648, 12015527936, 47438950400, 191117033472, 760625733632, 3082675150848, 12346305839104, 50223511928832, 202359539335168

Offset: 0

Gus Wiseman, Jul 07 2017

nonn

An enriched p-tree of weight n is either (case 1) the number n itself, or (case 2) a sequence of two or more enriched p-trees, having a weakly decreasing sequence of weights summing to n.

			The a(4) = 12 enriched p-trees are:
  4,
  (31), ((21)1), (((11)1)1), ((111)1),
  (22), (2(11)), ((11)2), ((11)(11)),
  (211), ((11)11),
  (1111).

Alois P. Heinz, Table of n, a(n) for n = 0..1588

Cf. A052337, A063834, A093637, A196545, A273873, A281145, A300660.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+a(i)*b(n-i, min(n-i, i))))
    end:
a:= n-> `if`(n=0, 1, 1+b(n, n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 07 2017

a[n_]:=a[n]=1+Sum[Times@@a/@y,{y,Rest[IntegerPartitions[n]]}];
Array[a,20]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1,
     If[i<1, 0, b[n, i-1] + a[i] b[n-i, Min[n-i, i]]]];
a[n_] := If[n == 0, 1, 1 + b[n, n-1]];
a /@ Range[0, 30] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

O.g.f.: (1/(1-x) + Product_{i>0} 1/(1-a(i)*x^i))/2.

A032305 Number of rooted trees where any 2 subtrees extending from the same node have a different number of nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 25, 51, 111, 240, 533, 1181, 2671, 6014, 13795, 31480, 72905, 168361, 393077, 914784, 2150810, 5040953, 11914240, 28089793, 66702160, 158013093, 376777192, 896262811, 2144279852, 5120176632, 12286984432, 29428496034, 70815501209

Offset: 1

Christian G. Bower

nonn

			The a(6) = 6 fully unbalanced trees: (((((o))))), (((o(o)))), ((o((o)))), (o(((o)))), (o(o(o))), ((o)((o))). - _Gus Wiseman_, Jan 10 2018

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Gus Wiseman, Illustration of the first 8 terms of A032305.
Index entries for sequences related to rooted trees

Cf. A000081, A001678, A003238, A004111, A213920, A273873, A290689, A291443, A297571.

Column k=1 of A318753.

A:= proc(n) if n<=1 then x else convert(series(x* (product(1+ coeff(A(n-1), x,i)*x^i, i=1..n-1)), x=0, n+1), polynom) fi end: a:= n-> coeff(A(n), x,n): seq(a(n), n=1..31);  # Alois P. Heinz, Aug 22 2008
# second Maple program:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(j=0, 1, g((i-1)$2))*g(n-i*j, i-1), j=0..min(1, n/i))))
    end:
a:= n-> g((n-1)$2):
seq(a(n), n=1..35);  # Alois P. Heinz, Mar 04 2013

nn=30;f[x_]:=Sum[a[n]x^n,{n,0,nn}];sol=SolveAlways[0 == Series[f[x]-x Product[1+a[i]x^i,{i,1,nn}],{x,0,nn}],x];Table[a[n],{n,1,nn}]/.sol  (* Geoffrey Critzer, Nov 17 2012 *)
allnim[n_]:=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[allnim/@c]],UnsameQ@@(Count[#,_List,{0,Infinity}]&/@#)&]]/@IntegerPartitions[n-1]];
Table[Length[allnim[n]],{n,15}] (* Gus Wiseman, Jan 10 2018 *)
g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[If[j == 0, 1, g[i-1, i-1]]*g[n-i*j, i-1], {j, 0, Min[1, n/i]}]]];
a[n_] := g[n-1, n-1];
Array[a, 35] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

a(n)=polcoeff(x*prod(i=1,n-1,1+a(i)*x^i)+x*O(x^n),n)

Shifts left under "EFK" (unordered, size, unlabeled) transform.
G.f.: A(x) = x*Product_{n>=1} (1+a(n)*x^n) = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Apr 07 2004
Lim_{n->infinity} a(n)^(1/n) = 2.5119824... - Vaclav Kotesovec, Nov 20 2019
G.f.: x * exp(Sum_{n>=1} Sum_{k>=1} (-1)^(k+1) * a(n)^k * x^(n*k) / k). - Ilya Gutkovskiy, Jun 30 2021

A299202 Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.

Original entry on oeis.org

0, 1, 1, -1, 1, -1, 1, 0, -1, -1, 1, 2, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, 1, -1, -1, -1, 0, 1, 1, 3, 1, 0, -1, -1, -1, -1, 1, -1, -1, -1, 1, 2, 1, 1, 1, -1, 1, 0, -1, 1, -1, 1, 1, -1, -1, -1, -1, -1, 1, -3, 1, -1, 2, 0, -1, 2, 1, 1, -1, 3, 1, 2, 1, -1, 1, 1, -1, 2, 1, 1, -1, -1, 1, -5, -1, -1, -1, -1, 1, -4

Offset: 1

Gus Wiseman, Feb 05 2018

sign

By convention, mu() = 0.

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

			Heinz number of (2,1,1) is 12, so mu(2,1,1) = a(12) = 2.

Gus Wiseman, Comcategories and Multiorders

Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299201, A299203.

nn=120;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
mu[y_]:=mu[y]=If[Length[y]===1,1,-Sum[Times@@mu/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y]&]}]];
mu/@ptns

mu(y) = Sum_{g(t)=y} (-1)^d(t), where the sum is over all enriched p-trees (A289501, A299203) whose multiset of leaves is the integer partition y, and d(t) is the number of non-leaf nodes in t.

A299200 Number of twice-partitions whose domain is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 15, 4, 22, 3, 10, 7, 30, 2, 9, 11, 8, 5, 42, 6, 56, 1, 14, 15, 15, 4, 77, 22, 22, 3, 101, 10, 135, 7, 12, 30, 176, 2, 25, 9, 30, 11, 231, 8, 21, 5, 44, 42, 297, 6, 385, 56, 20, 1, 33, 14, 490, 15, 60, 15, 627, 4

Offset: 1

Gus Wiseman, Feb 05 2018

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

			The a(15) = 6 twice-partitions: (3)(2), (3)(11), (21)(2), (21)(11), (111)(2), (111)(11).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299201, A299202, A299203.

Cf. A000720, A003964.

with(numtheory): with(combinat):
a:= n-> mul(numbpart(pi(i[1]))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..82);  # Alois P. Heinz, Jan 14 2021

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsP[PrimePi[p]]^k],{n,100}]

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = numbpart(primepi(f[k, 1]));); factorback(f);} \\ Michel Marcus, Feb 26 2018

Multiplicative with a(prime(n)) = A000041(n).

A281118 a(1)=1, a(n>1) = number of tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 2, 2, 12, 1, 6, 1, 6, 2, 2, 1, 20, 2, 2, 4, 6, 1, 8, 1, 32, 2, 2, 2, 28, 1, 2, 2, 20, 1, 8, 1, 6, 6, 2, 1, 76, 2, 6, 2, 6, 1, 20, 2, 20, 2, 2, 1, 38, 1, 2, 6, 112, 2, 8, 1, 6, 2, 8, 1, 116, 1, 2, 6, 6, 2, 8, 1, 76, 12, 2, 1

Offset: 1

Gus Wiseman, Jan 15 2017

nonn

A tree-factorization of n>=2 is either (case 1) the number n or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n. These are rooted plane trees and the ordering of branches is important. For example, {{2,2},9}, {2,{2,9}}, {{2,2},{3,3}}, {6,{2,3}}, and {{2,3},6} are distinct tree-factorizations of 36, but {9,{2,2}}, {{2,9},2}, and {{3,3},{2,2}} are not.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(30)=8 tree-factorizations are 30, 2*15, 2*(3*5), 3*10, 3*(2*5), 5*6, 5*(2*3), 2*3*5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001055, A063834, A196545, A262673, A273873, A281113, A289501.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
treefacs[n_]:=If[n<=1,{{}},Prepend[Join@@Function[q,Tuples[treefacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
Table[Length[treefacs[n]],{n,2,83}]

seq(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += w[k]^e*v[i]))); w} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A289501(n) for prime p. - Andrew Howroyd, Nov 18 2018

A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 4, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 4, 4, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 4, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Oct 25 2022

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. This sequence gives the number of ways to choose a strict partition of each prime index of n.

The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - Ivan N. Ianakiev, Oct 27 2022

			The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The non-strict version is A299200.
A horizontal version is A357978, non-strict A357977.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.
Cf. A003586, A059485.

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsQ[PrimePi[p]]^k],{n,100}]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f9(primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Oct 26 2022

A281145 Number of same-trees of weight n.

Original entry on oeis.org

1, 2, 2, 6, 2, 14, 2, 54, 10, 38, 2, 494, 2, 134, 42, 4470, 2, 3422, 2, 10262, 138, 2054, 2, 490926, 34, 8198, 1514, 314294, 2, 628318, 2, 30229110, 2058, 131078, 162, 150147342, 2, 524294, 8202, 628073814, 2, 109952254, 2, 371210294, 207370, 8388614, 2

Offset: 1

Gus Wiseman, Jan 15 2017

nonn

A same-tree is either: (case 1) a positive integer, or (case 2) a finite sequence of two or more same-trees all having the same weight, where the weight in case 2 is the sum of weights.

			The a(6)=14 same-trees are:
6,
(33),
(222),
(3(111)), ((111)3),
(22(11)), (2(11)2), ((11)22),
(2(11)(11)), ((11)2(11)), ((11)(11)2),
((111)(111)), ((11)(11)(11)), (111111).
The a(9)=10 same-trees are:
9,
(333),
(33(111)), (3(111)3), ((111)33),
(3(111)(111)), ((111)3(111)), ((111)(111)3),
((111)(111)(111)), (111111111).

G. C. Greubel, Table of n, a(n) for n = 1..4000

Cf. A196545, A260685, A273873, A275870, A006241.

a[n_]:=1+DivisorSum[n,b[#]^(n/#)&]-b[n]/.b->a;
Array[a,47]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sumdiv(n, d, v[n/d]^d)); v} \\ Andrew Howroyd, Aug 20 2018

a(n) = 1 + Sum a(d)^(n/d) where the sum is over divisors less than n.

A296122 Number of twice-partitions of n with no repeated partitions.

Original entry on oeis.org

1, 1, 2, 5, 10, 20, 40, 77, 157, 285, 552, 1018, 1921, 3484, 6436, 11622, 21082, 37550, 67681, 119318, 211792, 372003, 653496, 1137185, 1986234, 3429650, 5935970, 10205907, 17537684, 29958671, 51189932, 86967755, 147759421, 249850696, 422123392, 710495901

Offset: 0

Gus Wiseman, Dec 05 2017

nonn

a(n) is the number of sequences of distinct integer partitions whose sums are weakly decreasing and add up to n.

			The a(4) = 10 twice-partitions: (4), (31), (22), (211), (1111), (3)(1), (21)(1), (111)(1), (2)(11), (11)(2).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000009, A000041, A047968, A063834, A261049, A273873, A279375, A295279, A296121.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(j!*
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 06 2017

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@p],UnsameQ@@#&],{p,IntegerPartitions[n]}]],{n,15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[j!*
     Binomial[PartitionsP[i], j]*b[n - i*j, i - 1], {j, 0, n/i}]]];
a[n_] := b[n, n];
a /@ Range[0, 40] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(15)-a(34) from Robert G. Wilson v, Dec 06 2017

A299201 Number of twice-partitions whose composite is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 3, 4, 1, 6, 1, 7, 2, 2, 2, 11, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 16, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 13, 1, 2, 5, 11, 2, 5, 1, 4, 2, 6, 1, 19, 1, 2, 4, 4, 2, 5, 1, 13, 5, 2, 1, 13, 2

Offset: 1

Gus Wiseman, Feb 05 2018

nonn

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

			The a(36) = 11 twice-partitions:
  (2211),
  (22)(11), (211)(2), (221)(1), (21)(21),
  (2)(2)(11), (2)(11)(2), (11)(2)(2), (22)(1)(1), (21)(2)(1),
  (2)(2)(1)(1).

Robert Price, Table of n, a(n) for n = 1..200

Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299202, A299203.

nn=100;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
Table[Length[Select[tris,Sort[Join@@#,Greater]===y&]],{y,ptns}]

A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 28, 51, 92, 164, 289, 504, 871, 1493, 2539, 4290, 7201, 12017, 19939, 32911, 54044, 88330, 143709, 232817, 375640, 603755, 966816, 1542776, 2453536, 3889338, 6146126, 9683279, 15211881, 23830271, 37230720, 58015116, 90174847, 139820368, 216286593

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

Also the number of integer compositions of n whose leaders of maximal weakly decreasing runs are strictly increasing. For example, the composition (1,2,2,1,3,1,4,1) has maximal weakly decreasing runs ((1),(2,2,1),(3,1),(4,1)), with leaders (1,2,3,4), so is counted under a(15). - Gus Wiseman, Aug 21 2024

			The a(1) = 1 through a(5) = 15 multiset partitions:
  {1}  {2}    {3}        {4}          {5}
       {1,1}  {1,2}      {1,3}        {1,4}
              {1,1,1}    {2,2}        {2,3}
              {1},{1,1}  {1,1,2}      {1,1,3}
                         {1,1,1,1}    {1,2,2}
                         {1},{1,2}    {1,1,1,2}
                         {2},{1,1}    {1},{1,3}
                         {1},{1,1,1}  {1},{2,2}
                                      {2},{1,2}
                                      {3},{1,1}
                                      {1,1,1,1,1}
                                      {1},{1,1,2}
                                      {2},{1,1,1}
                                      {1},{1,1,1,1}
                                      {1,1},{1,1,1}
From _Gus Wiseman_, Aug 21 2024: (Start)
The a(0) = 1 through a(5) = 15 compositions whose leaders of maximal weakly decreasing runs are strictly increasing:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The version for set partitions is A007837.
For sums instead of sizes we have A271619.
For constant instead of distinct sizes we have A319066.
These multiset partitions are ranked by A326533.
For odd instead of distinct sizes we have A356932.
The version for twice-partitions is A358830.
The case of distinct sums also is A358832.
Ranked by positions of strictly increasing rows in A374740, opposite A374629.
A001970 counts multiset partitions of integer partitions.
A011782 counts compositions.
A063834 counts twice-partitions, strict A296122.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Runs and leaders: A000041, A188900, A374634, A374635, A374637, A374679, A374742 (ranks A374744), A374743 (ranks A374701), A374746, A374747.
Cf. A000009, A141199, A273873, A279375, A279785, A279790, A358334.
Cf. A003242, A106356, A188920, A189076, A238343, A333213, A336342.

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]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@Length/@#&]],{n,0,10}]
(* second program *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@First/@Split[#,GreaterEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 21 2024 *)

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, 1 + polcoef(g, k, y) + O(x*x^n)))} \\ Andrew Howroyd, Dec 31 2022

G.f.: Product_{k>=1} (1 + [y^k]P(x,y)) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2022

cp's OEIS Frontend

A289501 Number of enriched p-trees of weight n.

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A032305 Number of rooted trees where any 2 subtrees extending from the same node have a different number of nodes.

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A299202 Moebius function of the multiorder of integer partitions indexed by their Heinz numbers.

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A299200 Number of twice-partitions whose domain is the integer partition with Heinz number n.

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A281118 a(1)=1, a(n>1) = number of tree-factorizations of n.

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A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

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A281145 Number of same-trees of weight n.

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