A330727 -id:A330727 - cp's OEIS frontend

A330935 Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 2, 1, 0, 1, 2, 0, 1, 0, 1, 1, 0, 1, 5, 5, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 1, 0, 1, 5, 8, 4, 0, 1, 0, 1, 0, 1, 0, 1, 7, 7, 1, 0, 1, 0, 1, 0, 1, 5, 5, 1, 0, 1

Offset: 1

Gus Wiseman, Jan 04 2020

This poset is equivalent to the poset of multiset partitions of the prime indices of n, ordered by refinement.

			Triangle begins:
   1:          16: 0 1 3 2    31: 1            46: 0 1
   2: 1        17: 1          32: 0 1 5 8 4    47: 1
   3: 1        18: 0 1 2      33: 0 1          48: 0 1 10 23 15
   4: 0 1      19: 1          34: 0 1          49: 0 1
   5: 1        20: 0 1 2      35: 0 1          50: 0 1 2
   6: 0 1      21: 0 1        36: 0 1 7 7      51: 0 1
   7: 1        22: 0 1        37: 1            52: 0 1 2
   8: 0 1 1    23: 1          38: 0 1          53: 1
   9: 0 1      24: 0 1 5 5    39: 0 1          54: 0 1 5 5
  10: 0 1      25: 0 1        40: 0 1 5 5      55: 0 1
  11: 1        26: 0 1        41: 1            56: 0 1 5 5
  12: 0 1 2    27: 0 1 1      42: 0 1 3        57: 0 1
  13: 1        28: 0 1 2      43: 1            58: 0 1
  14: 0 1      29: 1          44: 0 1 2        59: 1
  15: 0 1      30: 0 1 3      45: 0 1 2        60: 0 1 9 11
Row n = 48 counts the following chains (minimum and maximum not shown):
  ()  (6*8)      (2*3*8)->(6*8)       (2*2*2*6)->(2*4*6)->(6*8)
      (2*24)     (2*4*6)->(6*8)       (2*2*3*4)->(2*3*8)->(6*8)
      (3*16)     (2*3*8)->(2*24)      (2*2*3*4)->(2*4*6)->(6*8)
      (4*12)     (2*3*8)->(3*16)      (2*2*2*6)->(2*4*6)->(2*24)
      (2*3*8)    (2*4*6)->(2*24)      (2*2*2*6)->(2*4*6)->(4*12)
      (2*4*6)    (2*4*6)->(4*12)      (2*2*3*4)->(2*3*8)->(2*24)
      (3*4*4)    (3*4*4)->(3*16)      (2*2*3*4)->(2*3*8)->(3*16)
      (2*2*12)   (3*4*4)->(4*12)      (2*2*3*4)->(2*4*6)->(2*24)
      (2*2*2*6)  (2*2*12)->(2*24)     (2*2*3*4)->(2*4*6)->(4*12)
      (2*2*3*4)  (2*2*12)->(4*12)     (2*2*3*4)->(3*4*4)->(3*16)
                 (2*2*2*6)->(6*8)     (2*2*3*4)->(3*4*4)->(4*12)
                 (2*2*3*4)->(6*8)     (2*2*2*6)->(2*2*12)->(2*24)
                 (2*2*2*6)->(2*24)    (2*2*2*6)->(2*2*12)->(4*12)
                 (2*2*2*6)->(4*12)    (2*2*3*4)->(2*2*12)->(2*24)
                 (2*2*3*4)->(2*24)    (2*2*3*4)->(2*2*12)->(4*12)
                 (2*2*3*4)->(3*16)
                 (2*2*3*4)->(4*12)
                 (2*2*2*6)->(2*4*6)
                 (2*2*3*4)->(2*3*8)
                 (2*2*3*4)->(2*4*6)
                 (2*2*3*4)->(3*4*4)
                 (2*2*2*6)->(2*2*12)
                 (2*2*3*4)->(2*2*12)

Row lengths are A001222.
Row sums are A317176.
Column k = 1 is A010051.
Column k = 2 is A066247.
Column k = 3 is A330936.
Final terms of each row are A317145.
The version for set partitions is A008826, with row sums A005121.
The version for integer partitions is A330785, with row sums A213427.
Cf. A001055, A002846, A003238, A007716, A281118, A292504, A292505, A318812, A330665, A330727.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
upfacs[q_]:=Union[Sort/@Join@@@Tuples[facs/@q]];
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[upfacs[y],y],{y,facs[n]}],{n},First[facs[n]]],Length[#]==k-1&]],{n,100},{k,PrimeOmega[n]}]

T(2^n,k) = A330785(n,k).

T(n,1) + T(n,2) = 1.

A318846 Number of balanced reduced multisystems whose atoms cover an initial interval of positive integers with multiplicities equal to the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 4, 15, 11, 20, 21, 90, 51, 80, 32, 468, 166, 2910, 124, 521, 277, 20644, 266, 621, 1761, 1866, 841, 165874, 1374, 1484344, 436, 3797, 12741, 5383, 3108, 14653890, 103783, 31323, 2294, 158136988, 12419, 1852077284, 6382, 20786, 939131, 23394406084

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

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}.

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.

			The a(12) = 21 multisystems on {1,1,2,3} (commas elided):
  {1123}  {{1}{123}}  {{1}{1}{23}}  {{{1}}{{1}{23}}}
          {{2}{113}}  {{1}{2}{13}}  {{{23}}{{1}{1}}}
          {{3}{112}}  {{1}{3}{12}}  {{{1}}{{2}{13}}}
          {{11}{23}}  {{2}{3}{11}}  {{{2}}{{1}{13}}}
          {{12}{13}}                {{{13}}{{1}{2}}}
                                    {{{1}}{{3}{12}}}
                                    {{{3}}{{1}{12}}}
                                    {{{12}}{{1}{3}}}
                                    {{{2}}{{3}{11}}}
                                    {{{3}}{{2}{11}}}
                                    {{{11}}{{2}{3}}}

Row sums of A330727.
Cf. A001055, A002846, A005121, A181821, A213427, A281118, A281119, A305936, A318762, A318812, A318813.
Cf. A318847, A318848, A318849.

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

a(n) = A318812(A181821(n)).
a(prime(n)) = A318813(n).
a(2^n) = A005121(n).

Terminology corrected by Gus Wiseman, Jan 04 2020

More terms from Jinyuan Wang, Jun 26 2020

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

A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 7, 5, 5, 11, 16, 16, 27, 18, 61, 62, 272, 45, 123, 61, 1385, 105, 152, 272, 501, 211, 7936, 362

Offset: 1

Gus Wiseman, Dec 30 2019

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.

			The a(n) multisystems for n = 3, 6, 8, 9, 10, 12 (commas and outer brackets elided):
  11  {1}{12}  {1}{23}  {{1}}{{1}{22}}  {{1}}{{1}{12}}  {{1}}{{1}{23}}
      {2}{11}  {2}{13}  {{11}}{{2}{2}}  {{11}}{{1}{2}}  {{11}}{{2}{3}}
               {3}{12}  {{1}}{{2}{12}}  {{1}}{{2}{11}}  {{1}}{{2}{13}}
                        {{12}}{{1}{2}}  {{12}}{{1}{1}}  {{12}}{{1}{3}}
                        {{2}}{{1}{12}}  {{2}}{{1}{11}}  {{1}}{{3}{12}}
                        {{2}}{{2}{11}}                  {{13}}{{1}{2}}
                        {{22}}{{1}{1}}                  {{2}}{{1}{13}}
                                                        {{2}}{{3}{11}}
                                                        {{23}}{{1}{1}}
                                                        {{3}}{{1}{12}}
                                                        {{3}}{{2}{11}}

The version with distinct atoms is A006472.
The non-maximal version is A318846.
A tree version is A318848, with orderless version A318849.
The unlabeled version is A330664.
Final terms in each row of A330727.
See also A330675 (strongly normal), A330676 (normal), and A330726 (partition).
Cf. A000111, A001055, A005121, A292504, A292505, A317145, A330665, A330666.

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

a(2^n) = A006472(n).

a(prime(n)) = A000111(n - 1).

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

A330935 Irregular triangle read by rows where T(n,k) is the number of length-k chains from minimum to maximum in the poset of factorizations of n into factors > 1, ordered by refinement.

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A318846 Number of balanced reduced multisystems whose atoms cover an initial interval of positive integers with multiplicities equal to the prime indices of n.

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

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A330728 Number of balanced reduced multisystems of maximum depth whose degrees (atom multiplicities) are the prime indices of n.

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

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