A330667 -id:A330667 - cp's OEIS frontend

A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 7, 1, 1, 1, 5, 1, 3, 1, 2, 2, 1, 1, 16, 1, 2, 1, 2, 1, 5, 1, 5, 1, 1, 1, 11, 1, 1, 2, 16, 1, 3, 1, 2, 1, 3, 1, 27, 1, 1, 2, 2, 1, 3, 1, 16, 2, 1, 1, 11, 1

Offset: 1

Gus Wiseman, Dec 27 2019

nonn

First differs from A317145 at a(32) = 5, A317145(32) = 4.
A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
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.
Also series/singleton-reduced factorizations of n with Omega(n) levels of parentheses. See A001055, A050336, A050338, A050340, etc.

			The a(n) multisystems for n = 2, 6, 12, 24, 48:
  {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}  {{{{1}}},{{{1}},{{1},{1,2}}}}
              {{2},{1,1}}  {{{1,1}},{{1},{2}}}  {{{{1}}},{{{1,1}},{{1},{2}}}}
                           {{{1}},{{2},{1,1}}}  {{{{1},{1}}},{{{1}},{{1,2}}}}
                           {{{1,2}},{{1},{1}}}  {{{{1},{1,1}}},{{{1}},{{2}}}}
                           {{{2}},{{1},{1,1}}}  {{{{1,1}}},{{{1}},{{1},{2}}}}
                                                {{{{1}}},{{{1}},{{2},{1,1}}}}
                                                {{{{1}}},{{{1,2}},{{1},{1}}}}
                                                {{{{1},{1}}},{{{2}},{{1,1}}}}
                                                {{{{1},{1,2}}},{{{1}},{{1}}}}
                                                {{{{1,1}}},{{{2}},{{1},{1}}}}
                                                {{{{1}}},{{{2}},{{1},{1,1}}}}
                                                {{{{1},{2}}},{{{1}},{{1,1}}}}
                                                {{{{1,2}}},{{{1}},{{1},{1}}}}
                                                {{{{2}}},{{{1}},{{1},{1,1}}}}
                                                {{{{2}}},{{{1,1}},{{1},{1}}}}
                                                {{{{2},{1,1}}},{{{1}},{{1}}}}

The last nonzero term in row n of A330667 is a(n).
The chain version is A317145.
The non-maximal version is A318812.
Unlabeled versions are A330664 and A330663.
Other labeled versions are A330675 (strongly normal) and A330676 (normal).
Cf. A001055, A005121, A005804, A050336, A213427, A292505, A317144, A318849, A320160, A330474, A330475, A330679.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

a(2^n) = A000111(n - 1).

a(product of n distinct primes) = A006472(n).

A008826 Triangle of coefficients from fractional iteration of e^x - 1.

Original entry on oeis.org

1, 1, 3, 1, 13, 18, 1, 50, 205, 180, 1, 201, 1865, 4245, 2700, 1, 875, 16674, 74165, 114345, 56700, 1, 4138, 155477, 1208830, 3394790, 3919860, 1587600, 1, 21145, 1542699, 19800165, 90265560, 182184030, 167310360, 57153600, 1, 115973, 16385857, 335976195, 2338275240, 7342024200, 11471572350, 8719666200, 2571912000

Offset: 2

N. J. A. Sloane, Mar 15 1996

The triangle reflects the Jordan-decomposition of the matrix of Stirling numbers of the second kind. A display of the matrix formula can be found at the Helms link which also explains the generation rule for the A()-numbers in a different way. - Gottfried Helms Apr 19 2014
From Gus Wiseman, Jan 02 2020: (Start)
Also the number of balanced reduced multisystems with atoms {1..n} and depth k. A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem. For example, row n = 4 counts the following multisystems:
  {1,2,3,4}  {{1},{2,3,4}}    {{{1}},{{2},{3,4}}}
             {{1,2},{3,4}}    {{{1},{2}},{{3,4}}}
             {{1,2,3},{4}}    {{{1},{2,3}},{{4}}}
             {{1,2,4},{3}}    {{{1,2}},{{3},{4}}}
             {{1,3},{2,4}}    {{{1,2},{3}},{{4}}}
             {{1,3,4},{2}}    {{{1},{2,4}},{{3}}}
             {{1,4},{2,3}}    {{{1,2},{4}},{{3}}}
             {{1},{2},{3,4}}  {{{1}},{{3},{2,4}}}
             {{1},{2,3},{4}}  {{{1},{3}},{{2,4}}}
             {{1,2},{3},{4}}  {{{1,3}},{{2},{4}}}
             {{1},{2,4},{3}}  {{{1,3},{2}},{{4}}}
             {{1,3},{2},{4}}  {{{1},{3,4}},{{2}}}
             {{1,4},{2},{3}}  {{{1,3},{4}},{{2}}}
                              {{{1}},{{4},{2,3}}}
                              {{{1},{4}},{{2,3}}}
                              {{{1,4}},{{2},{3}}}
                              {{{1,4},{2}},{{3}}}
                              {{{1,4},{3}},{{2}}}
(End)
From Harry Richman, Mar 30 2023: (Start)
Equivalently, T(n,k) is the number of length-k chains from minimum to maximum in the lattice of set partitions of {1..n} ordered by refinement. For example, row n = 4 counts the following chains, leaving out the minimum {1|2|3|4} and maximum {1234}:
  (empty)  {12|3|4}  {12|3|4} < {123|4}
           {13|2|4}  {12|3|4} < {124|3}
           {14|2|3}  {12|3|4} < {12|34}
           {1|23|4}  {13|2|4} < {123|4}
           {1|24|3}  {13|2|4} < {134|2}
           {1|2|34}  {13|2|4} < {13|24}
           {123|4}   {14|2|3} < {124|3}
           {124|3}   {14|2|3} < {134|2}
           {134|2}   {14|2|3} < {14|23}
           {1|234}   {1|23|4} < {123|4}
           {12|34}   {1|23|4} < {1|234}
           {13|24}   {1|23|4} < {14|23}
           {14|23}   {1|24|3} < {124|3}
                     {1|24|3} < {1|234}
                     {1|24|3} < {13|24}
                     {1|2|34} < {134|2}
                     {1|2|34} < {1|234}
                     {1|2|34} < {12|34}
(End)
Also the number of cells of dimension k in the fine subdivision of the Bergman complex of the complete graph on n vertices. - Harry Richman, Mar 30 2023

			Triangle starts:
  1;
  1,    3;
  1,   13,     18;
  1,   50,    205,     180;
  1,  201,   1865,    4245,    2700;
  1,  875,  16674,   74165,  114345,   56700;
  1, 4138, 155477, 1208830, 3394790, 3919860, 1587600;
  ...
The f-vector of (the fine subdivision of) the Bergman complex of the complete graph K_3 is (1, 3). The f-vector of the Bergman complex of K_4 is (1, 13, 18). - _Harry Richman_, Mar 30 2023

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.

Alois P. Heinz, Rows n = 2..150, flattened (first 19 rows from Vincenzo Librandi)
Gottfried Helms, How this expression leads to the given sequence, MathOverflow.
Federico Ardila and Caroline J. Klivans, The Bergman complex of a matroid and phylogenetic trees, J. Combin. Theory Ser. B, 96 (2006), 38-49.

Row sums are A005121.
Alternating row sums are signed factorials A133942(n-1).
Column k = 2 is A008827.
Diagonal k = n - 1 is A006472.
Diagonal k = n - 2 is A059355.
Row n equals row 2^n of A330727.
Cf. A000110, A000111, A000258, A002846, A005121, A008277, A306186, A317176, A318813, A320154, A330667, A330679, A330784.

b:= proc(n) option remember; expand(`if`(n=1, 1,
      add(Stirling2(n, j)*b(j)*x, j=0..n-1)))
    end:
T:= (n, k)-> coeff(b(n), x, k):
seq(seq(T(n, k), k=1..n-1), n=2..10);  # Alois P. Heinz, Mar 31 2023

a[n_, x_] := Sum[ StirlingS2[n, k]*a[k, x]*x, {k, 0, n-1}]; a[1, ] = 1; Table[ CoefficientList[ a[n, x], x] // Rest, {n, 2, 10}] // Flatten (* _Jean-François Alcover, Dec 11 2012, after Vladeta Jovovic *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
tots[m_]:=Prepend[Join@@Table[tots[p],{p,Select[sps[m],1Gus Wiseman, Jan 02 2020 *)

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} Stirling2(n, k)*A(k;x)*x, A(1;x) = 1. - Vladeta Jovovic, Jan 02 2004

Sum_{k=1..n-1} (-1)^k*T(n,k) = (-1)^(n-1)*(n-1)! = A133942(n-1). - Geoffrey Critzer, Sep 06 2020

More terms from Vladeta Jovovic, Jan 02 2004

A330727 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 1, 3, 1, 7, 7, 1, 5, 5, 1, 5, 9, 5, 1, 9, 11, 1, 9, 28, 36, 16, 1, 10, 24, 16, 1, 14, 38, 27, 1, 13, 18, 1, 13, 69, 160, 164, 61, 1, 24, 79, 62, 1, 20, 160, 580, 1022, 855, 272, 1, 19, 59, 45, 1, 27, 138, 232, 123, 1, 17, 77, 121, 61

Offset: 2

Gus Wiseman, Jan 04 2020

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

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. A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			Triangle begins:
   {}
   1
   1
   1   1
   1   2
   1   3   2
   1   3
   1   7   7
   1   5   5
   1   5   9   5
   1   9  11
   1   9  28  36  16
   1  10  24  16
   1  14  38  27
   1  13  18
   1  13  69 160 164  61
   1  24  79  62
For example, row n = 12 counts the following multisystems:
  {1,1,2,3}  {{1},{1,2,3}}    {{{1}},{{1},{2,3}}}
             {{1,1},{2,3}}    {{{1,1}},{{2},{3}}}
             {{1,2},{1,3}}    {{{1}},{{2},{1,3}}}
             {{2},{1,1,3}}    {{{1,2}},{{1},{3}}}
             {{3},{1,1,2}}    {{{1}},{{3},{1,2}}}
             {{1},{1},{2,3}}  {{{1,3}},{{1},{2}}}
             {{1},{2},{1,3}}  {{{2}},{{1},{1,3}}}
             {{1},{3},{1,2}}  {{{2}},{{3},{1,1}}}
             {{2},{3},{1,1}}  {{{2,3}},{{1},{1}}}
                              {{{3}},{{1},{1,2}}}
                              {{{3}},{{2},{1,1}}}

Row sums are A318846.
Final terms in each row are A330728.
Row prime(n) is row n of A330784.
Row 2^n is row n of A008826.
Row n is row A181821(n) of A330667.
Column k = 3 is A318284(n) - 2 for n > 2.
Cf. A000111, A002846, A005121, A292504, A318812, A318813, A318847, A318848, A318849, A330475, A330666, A330935.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[Reverse[FactorInteger[n]],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

T(2^n,k) = A008826(n,k).

A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 9, 28, 36, 16, 1, 13, 69, 160, 164, 61, 1, 20, 160, 580, 1022, 855, 272, 1, 28, 337, 1837, 4996, 7072, 4988, 1385

Offset: 2

Gus Wiseman, Jan 03 2020

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

			Triangle begins:
    1
    1    1
    1    3    2
    1    5    9    5
    1    9   28   36   16
    1   13   69  160  164   61
    1   20  160  580 1022  855  272
    1   28  337 1837 4996 7072 4988 1385
Row n = 5 counts the following multisystems (strings of 1's are replaced by their lengths):
  5  {1,4}      {{1},{1,3}}      {{{1}},{{1},{1,2}}}
     {2,3}      {{1},{2,2}}      {{{1,1}},{{1},{2}}}
     {1,1,3}    {{2},{1,2}}      {{{1}},{{2},{1,1}}}
     {1,2,2}    {{3},{1,1}}      {{{1,2}},{{1},{1}}}
     {1,1,1,2}  {{1},{1,1,2}}    {{{2}},{{1},{1,1}}}
                {{1,1},{1,2}}
                {{2},{1,1,1}}
                {{1},{1},{1,2}}
                {{1},{2},{1,1}}

Row sums are A318813.
Column k = 3 is A007042.
Column k = 4 is A001970(n) - 3*A000041(n) + 3.
Column k = n is A000111.
Row n is row prime(n) of A330727.
Cf. A000669, A001055, A002846, A005121, A196545, A213427, A318812, A320160, A330474, A330475, A330655, A330667, 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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

T(n,3) = A000041(n) - 2.

T(n,4) = A001970(n) - 3 * A000041(n) + 3.

cp's OEIS Frontend

A330665 Number of balanced reduced multisystems of maximal depth whose atoms are the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A008826 Triangle of coefficients from fractional iteration of e^x - 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A330727 Irregular triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k whose degrees (atom multiplicities) are the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula