A321717 -id:A321717 - cp's OEIS frontend

A321733 Number of (0,1)-matrices with n ones, no zero rows or columns, and the same row sums as column sums.

1, 1, 2, 8, 40, 246, 1816, 15630, 153592, 1696760, 20816358, 280807868, 4131117440, 65823490088, 1129256780408

Offset: 0

Gus Wiseman, Nov 18 2018

			The a(4) = 40 matrices:
  [1 1]
  [1 1]
.
  [1 1 0][1 1 0][1 0 1][1 0 1][1 0 0]
  [1 0 0][0 0 1][1 0 0][0 1 0][0 1 1]
  [0 0 1][1 0 0][0 1 0][1 0 0][0 1 0]
.
  [1 0 0][0 1 1][0 1 0][0 1 0][0 1 0]
  [0 0 1][1 0 0][1 1 0][1 0 1][0 1 1]
  [0 1 1][1 0 0][0 0 1][0 1 0][1 0 0]
.
  [0 1 0][0 0 1][0 0 1][0 0 1][0 0 1]
  [0 0 1][1 1 0][1 0 0][0 1 0][0 0 1]
  [1 0 1][0 1 0][0 1 1][1 0 1][1 1 0]
.
  [1 0 0 0][1 0 0 0][1 0 0 0][1 0 0 0][1 0 0 0][1 0 0 0]
  [0 1 0 0][0 1 0 0][0 0 1 0][0 0 1 0][0 0 0 1][0 0 0 1]
  [0 0 1 0][0 0 0 1][0 1 0 0][0 0 0 1][0 1 0 0][0 0 1 0]
  [0 0 0 1][0 0 1 0][0 0 0 1][0 1 0 0][0 0 1 0][0 1 0 0]
.
  [0 1 0 0][0 1 0 0][0 1 0 0][0 1 0 0][0 1 0 0][0 1 0 0]
  [1 0 0 0][1 0 0 0][0 0 1 0][0 0 1 0][0 0 0 1][0 0 0 1]
  [0 0 1 0][0 0 0 1][1 0 0 0][0 0 0 1][1 0 0 0][0 0 1 0]
  [0 0 0 1][0 0 1 0][0 0 0 1][1 0 0 0][0 0 1 0][1 0 0 0]
.
  [0 0 1 0][0 0 1 0][0 0 1 0][0 0 1 0][0 0 1 0][0 0 1 0]
  [1 0 0 0][1 0 0 0][0 1 0 0][0 1 0 0][0 0 0 1][0 0 0 1]
  [0 1 0 0][0 0 0 1][1 0 0 0][0 0 0 1][1 0 0 0][0 1 0 0]
  [0 0 0 1][0 1 0 0][0 0 0 1][1 0 0 0][0 1 0 0][1 0 0 0]
.
  [0 0 0 1][0 0 0 1][0 0 0 1][0 0 0 1][0 0 0 1][0 0 0 1]
  [1 0 0 0][1 0 0 0][0 1 0 0][0 1 0 0][0 0 1 0][0 0 1 0]
  [0 1 0 0][0 0 1 0][1 0 0 0][0 0 1 0][1 0 0 0][0 1 0 0]
  [0 0 1 0][0 1 0 0][0 0 1 0][1 0 0 0][0 1 0 0][1 0 0 0]

Cf. A006052, A007016, A049311, A054976, A057151, A104602, A120732, A319056, A321717, A321723, A321732, A321735, A321736, A321739.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#]==Union[Last/@#],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]

a(7)-a(14) from Lars Blomberg, May 23 2019

A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 3, 2, 2, 4, 2, 3, 3, 4, 4, 4, 3, 5, 4, 5, 6, 6, 6, 6, 6, 8, 6, 7, 9, 8, 11, 8, 11, 11, 11, 12, 13, 13, 15, 13, 17, 17, 18, 18, 17, 20, 22, 21, 24, 24, 24, 26, 29, 28, 33, 30, 35, 34, 38, 38, 45, 42, 43, 45, 48, 52, 54, 55, 59, 59, 65, 65, 72, 73

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

Such a partition must be strict (unless it is all 1's) and its parts must also be squarefree.

			The a(30) = 8 integer partitions:
  (30),
  (17,13),(19,11),(23,7),
  (17,11,2),(23,5,2),
  (13,7,5,3,2),
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).

Lucas A. Brown, Table of n, a(n) for n = 0..133
Lucas A. Brown, Python program.

Cf. A002865, A003963, A005117, A038041, A064573, A073576, A302505, A319056, A320322, A321717, A321718, A322526, A322527, A322528, A322530, A322531.

Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SquareFreeQ[Times@@#]]&]],{n,30}]

a(51)-a(69) from Jinyuan Wang, Jun 27 2020

a(70) onwards from Lucas A. Brown, Aug 17 2024

A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 9, 13, 15, 16, 19, 27, 32, 49, 53, 64, 81, 113, 128, 131, 151, 161, 165, 169, 225, 243, 256, 311, 343, 361, 512, 719, 729, 1024, 1291, 1321, 1619, 1937, 1957, 2021, 2048, 2093, 2117, 2187, 2197, 2257, 2401, 2805, 2809, 3375, 3671, 4096, 6561

Offset: 1

Gus Wiseman, Dec 27 2018

nonn

A multiset multisystem is a finite multiset of finite multisets. 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}}.

A multiset multisystem is uniform if all parts have the same size, and regular if all vertices appear the same number of times. For example, {{1,1},{2,3},{2,3}} is uniform, regular, and spans an initial interval of positive integers, so its MM-number 15463 belongs to the sequence.

			The sequence of all uniform regular multiset multisystems spanning an initial interval of positive integers, together with their MM-numbers, begins:
    1: {}
    2: {{}}
    3: {{1}}
    4: {{},{}}
    7: {{1,1}}
    8: {{},{},{}}
    9: {{1},{1}}
   13: {{1,2}}
   15: {{1},{2}}
   16: {{},{},{},{}}
   19: {{1,1,1}}
   27: {{1},{1},{1}}
   32: {{},{},{},{},{}}
   49: {{1,1},{1,1}}
   53: {{1,1,1,1}}
   64: {{},{},{},{},{},{}}
   81: {{1},{1},{1},{1}}
  113: {{1,2,3}}
  128: {{},{},{},{},{},{},{}}
  131: {{1,1,1,1,1}}
  151: {{1,1,2,2}}
  161: {{1,1},{2,2}}
  165: {{1},{2},{3}}
  169: {{1,2},{1,2}}
  225: {{1},{1},{2},{2}}
  243: {{1},{1},{1},{1},{1}}
  256: {{},{},{},{},{},{},{},{}}
  311: {{1,1,1,1,1,1}}
  343: {{1,1},{1,1},{1,1}}
  361: {{1,1,1},{1,1,1}}
  512: {{},{},{},{},{},{},{},{},{}}
  719: {{1,1,1,1,1,1,1}}
  729: {{1},{1},{1},{1},{1},{1}}

Cf. A005176, A007016, A112798, A302242, A306021, A319056, A319189, A320324, A321698, A321717, A322554, A322703, A322833.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[1000],And[normQ[primeMS/@primeMS[#]],SameQ@@PrimeOmega/@primeMS[#],SameQ@@Last/@FactorInteger[Times@@primeMS[#]]]&]

A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 13, 15, 16, 17, 29, 31, 32, 33, 41, 43, 47, 51, 55, 59, 64, 67, 73, 79, 83, 85, 93, 101, 109, 113, 123, 127, 128, 137, 139, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 199, 201, 205, 211, 233, 241, 249, 255, 256, 257, 269, 271

Offset: 1

Gus Wiseman, Dec 14 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
All entries are themselves squarefree numbers (except the powers of 2).
The first odd term not in this sequence but in A302521 is 141, which is the MM-number (see A302242) of {{1},{2,3}}.

			The sequence of all integer partitions whose parts all have the same number of prime factors and whose product of parts is a squarefree number begins: (), (1), (2), (1,1), (3), (1,1,1), (5), (6), (3,2), (1,1,1,1), (7), (10), (11), (1,1,1,1,1), (5,2), (13), (14), (15), (7,2), (5,3), (17), (1,1,1,1,1,1).

Cf. A003963, A005117, A038041, A056239, A073576, A112798, A302242, A302505, A306017, A319056, A319169, A320324, A321717, A321718, A322526, A322528.

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

cp's OEIS Frontend

A321733 Number of (0,1)-matrices with n ones, no zero rows or columns, and the same row sums as column sums.

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A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

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A321699 MM-numbers of uniform regular multiset multisystems spanning an initial interval of positive integers.

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A322531 Heinz numbers of integer partitions whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

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Mathematica