cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A322785 Number of uniform multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

Original entry on oeis.org

1, 1, 4, 4, 12, 4, 48, 4, 183, 297, 1186, 4, 33950, 4, 139527, 1529608, 4726356, 4, 229255536, 4, 3705777010, 36279746314, 13764663019, 4, 14096735197959, 5194673049514, 7907992957755, 2977586461058927, 13426396910491001, 4, 1350012288268171854, 4, 59487352224070807287
Offset: 0

Views

Author

Gus Wiseman, Dec 26 2018

Keywords

Comments

A multiset is uniform if all multiplicities are equal. A multiset partition is uniform if all parts have the same size.

Examples

			The a(1) = 1 though a(6) = 48 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}          {111111}
       {12}    {123}      {1122}        {12345}          {111222}
       {1}{1}  {1}{1}{1}  {1234}        {1}{1}{1}{1}{1}  {112233}
       {1}{2}  {1}{2}{3}  {11}{11}      {1}{2}{3}{4}{5}  {123456}
                          {11}{22}                       {111}{111}
                          {12}{12}                       {111}{222}
                          {12}{34}                       {112}{122}
                          {13}{24}                       {112}{233}
                          {14}{23}                       {113}{223}
                          {1}{1}{1}{1}                   {122}{133}
                          {1}{1}{2}{2}                   {123}{123}
                          {1}{2}{3}{4}                   {123}{456}
                                                         {124}{356}
                                                         {125}{346}
                                                         {126}{345}
                                                         {134}{256}
                                                         {135}{246}
                                                         {136}{245}
                                                         {145}{236}
                                                         {146}{235}
                                                         {156}{234}
                                                         {11}{11}{11}
                                                         {11}{12}{22}
                                                         {11}{22}{33}
                                                         {11}{23}{23}
                                                         {12}{12}{12}
                                                         {12}{12}{33}
                                                         {12}{13}{23}
                                                         {12}{34}{56}
                                                         {12}{35}{46}
                                                         {12}{36}{45}
                                                         {13}{13}{22}
                                                         {13}{24}{56}
                                                         {13}{25}{46}
                                                         {13}{26}{45}
                                                         {14}{23}{56}
                                                         {14}{25}{36}
                                                         {14}{26}{35}
                                                         {15}{23}{46}
                                                         {15}{24}{36}
                                                         {15}{26}{34}
                                                         {16}{23}{45}
                                                         {16}{24}{35}
                                                         {16}{25}{34}
                                                         {1}{1}{1}{1}{1}{1}
                                                         {1}{1}{1}{2}{2}{2}
                                                         {1}{1}{2}{2}{3}{3}
                                                         {1}{2}{3}{4}{5}{6}

Links

Crossrefs

Row sums of A322788.
Cf. A000040, A038041, A072774, A100778, A299353, A306017, A306018, A306021, A317583, A317584, A319056, A319189, A321721, A322705, A322784, A322788.

Programs

  • Mathematica
    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[Sum[Length[Select[mps[m],SameQ@@Length/@#&]],{m,Table[Join@@Table[Range[n/d],{d}],{d,Divisors[n]}]}],{n,8}]

Formula

a(n) = 4 <=> n in { A000040 }. - Alois P. Heinz, Feb 03 2022

Extensions

More terms from Alois P. Heinz, Jan 30 2019
Terms a(14) and beyond from Andrew Howroyd, Feb 03 2022

A322786 Irregular triangle read by rows where T(n,k) is the number of multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 5, 3, 15, 9, 5, 52, 7, 203, 66, 31, 11, 877, 15, 4140, 712, 109, 22, 21147, 686, 30, 115975, 10457, 339, 42, 678570, 56, 4213597, 198091, 27036, 6721, 1043, 77, 27644437, 101, 190899322, 4659138, 2998, 135, 1382958545, 1688360, 58616, 176
Offset: 1

Views

Author

Gus Wiseman, Dec 26 2018

Keywords

Examples

			Triangle begins:
        1
        2       2
        5       3
       15       9       5
       52       7
      203      66      31      11
      877      15
     4140     712     109      22
    21147     686      30
   115975   10457     339      42
   678570      56
  4213597  198091   27036    6721    1043      77
For example, row 4 counts the following multiset partitions.
  {{1,2,3,4}}        {{1,1,2,2}}        {{1,1,1,1}}
  {{1},{2,3,4}}      {{1},{1,2,2}}      {{1},{1,1,1}}
  {{1,2},{3,4}}      {{1,1},{2,2}}      {{1,1},{1,1}}
  {{1,3},{2,4}}      {{1,2},{1,2}}      {{1},{1},{1,1}}
  {{1,4},{2,3}}      {{2},{1,1,2}}      {{1},{1},{1},{1}}
  {{2},{1,3,4}}      {{1},{1},{2,2}}
  {{3},{1,2,4}}      {{1},{2},{1,2}}
  {{4},{1,2,3}}      {{2},{2},{1,1}}
  {{1},{2},{3,4}}    {{1},{1},{2},{2}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{4},{1,2}}
  {{1},{2},{3},{4}}

Links

Crossrefs

Row sums are A322784. First column is A000110.
Cf. A001055, A005176, A027750, A056239, A072774, A100778, A219727, A295193, A306017, A319190, A319612, A322784, A322785, A322787, A322788, A322792.

Programs

  • Mathematica
    u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
Table[Table[u[Array[Prime,n/d,1,Times]^d,Array[Prime,n/d,1,Times]^d],{d,Divisors[n]}],{n,10}]
  • PARI
    \\ needs T(n,k) from A219727.
Row(n)={[T(d,n/d) | d<-divisors(n)]}
{ for(n=1, 12, print(Row(n))) } \\ Andrew Howroyd, Jan 11 2020

Formula

T(n,k) = A001055(A002110(n/d)^d), where d = A027750(n,k).
T(n,k) = A219727(d, n/d), where d = A027750(n, k). - Andrew Howroyd, Jan 11 2020

Extensions

Edited by Peter Munn, Mar 05 2025

A322787 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic multiset partitions of a multiset with d = A027750(n, k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 7, 5, 7, 7, 11, 23, 21, 11, 15, 15, 22, 79, 66, 22, 30, 162, 30, 42, 274, 192, 42, 56, 56, 77, 1003, 1636, 1338, 565, 77, 101, 101, 135, 3763, 1579, 135, 176, 19977, 10585, 176, 231, 14723, 43686, 4348, 231, 297, 297, 385, 59663, 298416, 82694, 11582, 385
Offset: 1

Views

Author

Gus Wiseman, Dec 26 2018

Keywords

Examples

			Triangle begins:
   1
   2   2
   3   3
   5   7   5
   7   7
  11  23  21  11
  15  15
  22  79  66  22
  30 162  30
  42 274 192  42
Non-isomorphic representatives of the multiset partitions counted under row 6:
{123456}           {112233}           {111222}           {111111}
{1}{23456}         {1}{12233}         {1}{11222}         {1}{11111}
{12}{3456}         {11}{2233}         {11}{1222}         {11}{1111}
{123}{456}         {112}{233}         {111}{222}         {111}{111}
{1}{2}{3456}       {12}{1233}         {112}{122}         {1}{1}{1111}
{1}{23}{456}       {123}{123}         {12}{1122}         {1}{11}{111}
{12}{34}{56}       {1}{1}{2233}       {1}{1}{1222}       {11}{11}{11}
{1}{2}{3}{456}     {1}{12}{233}       {1}{11}{222}       {1}{1}{1}{111}
{1}{2}{34}{56}     {11}{22}{33}       {11}{12}{22}       {1}{1}{11}{11}
{1}{2}{3}{4}{56}   {11}{23}{23}       {1}{12}{122}       {1}{1}{1}{1}{11}
{1}{2}{3}{4}{5}{6} {1}{2}{1233}       {1}{2}{1122}       {1}{1}{1}{1}{1}{1}
                   {12}{13}{23}       {12}{12}{12}
                   {1}{23}{123}       {2}{11}{122}
                   {2}{11}{233}       {1}{1}{1}{222}
                   {1}{1}{2}{233}     {1}{1}{12}{22}
                   {1}{1}{22}{33}     {1}{1}{2}{122}
                   {1}{1}{23}{23}     {1}{2}{11}{22}
                   {1}{2}{12}{33}     {1}{2}{12}{12}
                   {1}{2}{13}{23}     {1}{1}{1}{2}{22}
                   {1}{2}{3}{123}     {1}{1}{2}{2}{12}
                   {1}{1}{2}{2}{33}   {1}{1}{1}{2}{2}{2}
                   {1}{1}{2}{3}{23}
                   {1}{1}{2}{2}{3}{3}

Links

Crossrefs

Row sums are A306017. First column is A000041.
Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306018, A318951, A319190, A319612, A322784, A322785, A322788, A322792.

Programs

  • PARI
    \\ See A318951 for RowSumMats
row(n)={my(d=divisors(n)); vector(#d, i, RowSumMats(n/d[i], n, d[i]))}
{ for(n=1, 15, print(row(n))) } \\ Andrew Howroyd, Feb 02 2022

Extensions

Terms a(28) and beyond from Andrew Howroyd, Feb 02 2022
Name edited by Peter Munn, Mar 05 2025
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.