A005176 -id:A005176 - cp's OEIS frontend

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.

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

Gus Wiseman, Dec 26 2018

			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}}

Andrew Howroyd, Table of n, a(n) for n = 1..207 (first 50 rows)

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.

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}]

\\ 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

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

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

Gus Wiseman, Dec 26 2018

			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}

Andrew Howroyd, Table of n, a(n) for n = 1..207 (rows 1..50)

Row sums are A306017. First column is A000041.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306018, A318951, A319190, A319612, A322784, A322785, A322788, A322792.

\\ 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

Terms a(28) and beyond from Andrew Howroyd, Feb 02 2022

Name edited by Peter Munn, Mar 05 2025

A322789 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic uniform 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, 2, 2, 3, 4, 3, 2, 2, 4, 7, 6, 4, 2, 2, 4, 10, 8, 4, 3, 7, 3, 4, 12, 8, 4, 2, 2, 6, 32, 35, 31, 18, 6, 2, 2, 4, 21, 10, 4, 4, 47, 29, 4, 5, 49, 72, 19, 5, 2, 2, 6, 81, 170, 71, 24, 6, 2, 2, 6, 138, 478, 296, 32, 6, 4, 429, 76, 4, 4, 64, 14, 4

Offset: 1

Gus Wiseman, Dec 26 2018

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

			Triangle begins:
  1
  2  2
  2  2
  3  4  3
  2  2
  4  7  6  4
  2  2
  4 10  8  4
  3  7  3
  4 12  8  4
Non-isomorphic representatives of the multiset partitions counted under row 6:
{123456}           {112233}           {111222}           {111111}
{123}{456}         {112}{233}         {111}{222}         {111}{111}
{12}{34}{56}       {123}{123}         {112}{122}         {11}{11}{11}
{1}{2}{3}{4}{5}{6} {11}{22}{33}       {11}{12}{22}       {1}{1}{1}{1}{1}{1}
                   {11}{23}{23}       {12}{12}{12}
                   {12}{13}{23}       {1}{1}{1}{2}{2}{2}
                   {1}{1}{2}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 1..338 (rows 1..75)

Row sums are A319056. First column is A000005.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A306018, A319190, A319612, A322784, A322785, A322787, A322792.

Terms a(28) and beyond from Andrew Howroyd, Feb 03 2022

Name edited by Peter Munn, Mar 05 2025

A319877 Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

Original entry on oeis.org

1, 7, 9, 14, 18, 23, 25, 28, 36, 46, 50, 56, 72, 92, 97, 100, 112, 121, 144, 151, 161, 169, 175, 183, 184, 185, 194, 195, 200, 207, 224, 225, 227, 242, 288, 289, 302, 322, 338, 350, 366, 368, 370, 388, 390, 400, 414, 448, 450, 454, 484, 541, 576, 578, 604, 644

Offset: 1

Gus Wiseman, Dec 17 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. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of 2-regular multiset multisystems (meaning all vertex-degrees are 2).

			The sequence of multiset multisystems whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   14: {{},{1,1}}
   18: {{},{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   28: {{},{},{1,1}}
   36: {{},{},{1},{1}}
   46: {{},{2,2}}
   50: {{},{2},{2}}
   56: {{},{},{},{1,1}}
   72: {{},{},{},{1},{1}}
   92: {{},{},{2,2}}
   97: {{3,3}}
  100: {{},{},{2},{2}}
  112: {{},{},{},{},{1,1}}
  121: {{3},{3}}
  144: {{},{},{},{},{1},{1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  183: {{1},{1,2,2}}
  184: {{},{},{},{2,2}}
  185: {{2},{1,1,2}}
  194: {{},{3,3}}
  195: {{1},{2},{1,2}}
  200: {{},{},{},{2},{2}}

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319878, A319899, A320325, A322526, A322527, A322530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 472, 23342

Offset: 0

Gus Wiseman, Dec 23 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is k-uniform if all edges contain exactly k vertices, and k-regular if all vertices belong to exactly k edges. The span of a hypergraph is the union of its edges.

			The a(3) = 2 hypergraphs:
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
The a(4) = 5 hypergraphs:
  {{1},{2},{3},{4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The a(5) = 26 hypergraphs:
  {{1},{2},{3},{4},{5}}
  {{1,2},{1,3},{2,4},{3,5},{4,5}}
  {{1,2},{1,3},{2,5},{3,4},{4,5}}
  {{1,2},{1,4},{2,3},{3,5},{4,5}}
  {{1,2},{1,4},{2,5},{3,4},{3,5}}
  {{1,2},{1,5},{2,3},{3,4},{4,5}}
  {{1,2},{1,5},{2,4},{3,4},{3,5}}
  {{1,3},{1,4},{2,3},{2,5},{4,5}}
  {{1,3},{1,4},{2,4},{2,5},{3,5}}
  {{1,3},{1,5},{2,3},{2,4},{4,5}}
  {{1,3},{1,5},{2,4},{2,5},{3,4}}
  {{1,4},{1,5},{2,3},{2,4},{3,5}}
  {{1,4},{1,5},{2,3},{2,5},{3,4}}
  {{1,2,3},{1,2,4},{1,3,5},{2,4,5},{3,4,5}}
  {{1,2,3},{1,2,4},{1,4,5},{2,3,5},{3,4,5}}
  {{1,2,3},{1,2,5},{1,3,4},{2,4,5},{3,4,5}}
  {{1,2,3},{1,2,5},{1,4,5},{2,3,4},{3,4,5}}
  {{1,2,3},{1,3,4},{1,4,5},{2,3,5},{2,4,5}}
  {{1,2,3},{1,3,5},{1,4,5},{2,3,4},{2,4,5}}
  {{1,2,4},{1,2,5},{1,3,4},{2,3,5},{3,4,5}}
  {{1,2,4},{1,2,5},{1,3,5},{2,3,4},{3,4,5}}
  {{1,2,4},{1,3,4},{1,3,5},{2,3,5},{2,4,5}}
  {{1,2,4},{1,3,5},{1,4,5},{2,3,4},{2,3,5}}
  {{1,2,5},{1,3,4},{1,3,5},{2,3,4},{2,4,5}}
  {{1,2,5},{1,3,4},{1,4,5},{2,3,4},{2,3,5}}
  {{1,2,3,4},{1,2,3,5},{1,2,4,5},{1,3,4,5},{2,3,4,5}}

Row sums of A322706.

Cf. A005176, A058891, A059441, A116539, A295193, A299353, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{k}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,n}],{n,1,6}]

A350911 Number of regular digraphs on n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 56, 609, 39346, 8728237, 6126648298, 14487876826313

Offset: 0

Andrew Howroyd, Jan 29 2022

Row sums of A350910.

Cf. A000273, A005176, A329234, A350913.

A381586 Number of simple graphs on n unlabeled vertices whose degree sequence is consecutive.

Original entry on oeis.org

1, 1, 2, 4, 9, 24, 98, 622, 7293, 162052, 6997100, 578605618, 90558592724, 26673271109299, 14758661765740616

Offset: 0

John P. McSorley, Feb 28 2025

A graph has a consecutive degree sequence if its distinct degrees are consecutive integers. This includes all regular graphs.

			For n = 4 there are 11 non-isomorphic graphs G on 4 vertices. An example with consecutive degree sequence is 4K_1, with degree sequence 0000; and an example with non-consecutive degree sequence is K_1 U K_3 with degree sequence 0222. The only other G with non-consecutive degree sequence is K_{1,3} with degree sequence 1113. Thus a(4) = 9.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford University Press (1999).

Cf. A000088, A005176.

a(8)-a(14) from Andrew Howroyd, Feb 28 2025

A051427 Number of strictly Deza graphs with n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 2, 1, 0, 6, 1, 1, 1

Offset: 1

N. J. A. Sloane

From the Erikson et al. paper: We consider the following generalization of strongly regular graphs. A graph G is a Deza graph if it is regular and the number of common neighbors of two distinct vertices takes on one of two values (not necessarily depending on the adjacency of the two vertices). - Jonathan Vos Post, Jul 06 2008

M. Erickson, S. Fernando, W. H. Haemers, D. Hardy and J. Hemmeter, Deza graphs: A generalization of strongly regular graph, J. Combinatorial Designs, Vol 7, Issue 6, 395-405, Oct 21, 1999. See also here.
Sean A. Irvine, Java program (github)

Cf. A005176, A076434, A076435, A088741.

a(14)-a(15) from Sean A. Irvine, Sep 18 2021

A319878 Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

Original entry on oeis.org

1, 7, 9, 23, 25, 97, 121, 151, 161, 169, 175, 183, 185, 195, 207, 225, 227, 289, 541, 661, 679, 687, 781, 841, 847, 873, 957, 961, 1009, 1089, 1193, 1427, 1563, 1589, 1681, 1819, 1849, 1879, 1895, 2023, 2043, 2167, 2193, 2209, 2231, 2425, 2437, 2585, 2601

Offset: 1

Gus Wiseman, Dec 17 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. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. This sequence lists all MM-numbers of 2-regular (all vertex-degrees are 2) multiset partitions (no empty parts).

			The sequence of multiset partitions whose MM-numbers belong to the sequence begins:
    1: {}
    7: {{1,1}}
    9: {{1},{1}}
   23: {{2,2}}
   25: {{2},{2}}
   97: {{3,3}}
  121: {{3},{3}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  169: {{1,2},{1,2}}
  175: {{2},{2},{1,1}}
  183: {{1},{1,2,2}}
  185: {{2},{1,1,2}}
  195: {{1},{2},{1,2}}
  207: {{1},{1},{2,2}}
  225: {{1},{1},{2},{2}}
  227: {{4,4}}
  289: {{4},{4}}
  541: {{1,1,3,3}}
  661: {{5,5}}
  679: {{1,1},{3,3}}
  687: {{1},{1,3,3}}
  781: {{3},{1,1,3}}
  841: {{1,3},{1,3}}
  847: {{1,1},{3},{3}}
  873: {{1},{1},{3,3}}
  957: {{1},{3},{1,3}}
  961: {{5},{5}}

Cf. A003963, A005117, A005176, A062503, A064573, A072774, A295193, A302505, A319877, A319899, A320325, A322526, A322527, A322530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],Or[#==1,SameQ[##,2]&@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322555 Number of labeled simple graphs on n vertices where all non-isolated vertices have the same degree.

Original entry on oeis.org

1, 1, 2, 5, 18, 69, 390, 2703, 59474, 1548349, 168926258, 12165065351, 7074423247562, 2294426405580191, 4218009215702391954, 3810376434461484994317, 35102248193591661086921250, 156873334244228518638713087133, 4144940994226400702145709978234154

Offset: 0

Gus Wiseman, Dec 15 2018

nonn

Such graphs may be said to have regular support.

			The a(4) = 18 edge sets:
  {}
  {{1,2}}
  {{1,3}}
  {{1,4}}
  {{2,3}}
  {{2,4}}
  {{3,4}}
  {{1,2},{3,4}}
  {{1,3},{2,4}}
  {{1,4},{2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A000569, A005176, A038041, A059441, A295193, A306017, A306019, A319169, A320458.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],SameQ@@Length/@Split[Sort[Join@@#]]&]],{n,6}]

a(n) = 1 + Sum_{k=1..n} binomial(n, k)*(A295193(k) - 1). - Andrew Howroyd, Dec 17 2018

a(8)-a(15) from Andrew Howroyd, Dec 17 2018

a(16)-a(18) from Andrew Howroyd, May 21 2020

cp's OEIS Frontend

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.

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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.

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A322789 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

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A319877 Numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

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A322705 Number of k-uniform k-regular hypergraphs spanning n vertices, for some 1 <= k <= n.

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A350911 Number of regular digraphs on n unlabeled nodes.

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A381586 Number of simple graphs on n unlabeled vertices whose degree sequence is consecutive.

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A051427 Number of strictly Deza graphs with n nodes.

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A319878 Odd numbers whose product of prime indices (A003963) is a square of a squarefree number (A062503).

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