A323351 -id:A323351 - cp's OEIS frontend

A323300 Number of ways to fill a matrix with the parts of the integer partition with Heinz number n.

1, 1, 1, 2, 1, 4, 1, 2, 2, 4, 1, 6, 1, 4, 4, 3, 1, 6, 1, 6, 4, 4, 1, 12, 2, 4, 2, 6, 1, 12, 1, 2, 4, 4, 4, 18, 1, 4, 4, 12, 1, 12, 1, 6, 6, 4, 1, 10, 2, 6, 4, 6, 1, 12, 4, 12, 4, 4, 1, 36, 1, 4, 6, 4, 4, 12, 1, 6, 4, 12, 1, 20, 1, 4, 6, 6, 4, 12, 1, 10, 3, 4

Offset: 1

Gus Wiseman, Jan 12 2019

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(24) = 12 matrices whose entries are (2,1,1,1):
  [1 1 1 2] [1 1 2 1] [1 2 1 1] [2 1 1 1]
.
  [1 1] [1 1] [1 2] [2 1]
  [1 2] [2 1] [1 1] [1 1]
.
  [1] [1] [1] [2]
  [1] [1] [2] [1]
  [1] [2] [1] [1]
  [2] [1] [1] [1]

Positions of 1's are one and prime numbers A008578.
Positions of 2's are primes to prime powers A053810.
Cf. A000005, A001222, A008480, A056239, A063989, A112798, A120733.
Cf. A323295, A323305, A323307, A323351.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Array[Length[ptnmats[#]]&,100]

a(n) = A008480(n) * A000005(A001222(n)).

A323865 Number of aperiodic binary toroidal necklaces of size n.

Original entry on oeis.org

1, 2, 2, 4, 8, 12, 36, 36, 114, 166, 396, 372, 1992, 1260, 4644, 8728, 20310, 15420, 87174, 55188, 314064, 399432, 762228, 729444, 5589620, 4026522, 10323180, 19883920, 57516048, 37025580, 286322136, 138547332, 805277760, 1041203944, 2021145660, 3926827224

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			Inequivalent representatives of the a(6) = 36 aperiodic necklaces:
  000001  000011  000101  000111  001011  001101  001111  010111  011111
.
  000  000  001  001  001  001  001  011  011
  001  011  010  011  101  110  111  101  111
.
  00  00  00  00  00  01  01  01  01
  00  01  01  01  11  01  01  10  11
  01  01  10  11  01  10  11  11  11
.
  0  0  0  0  0  0  0  0  0
  0  0  0  0  0  0  0  1  1
  0  0  0  0  1  1  1  0  1
  0  0  1  1  0  1  1  1  1
  0  1  0  1  1  0  1  1  1
  1  1  1  1  1  1  1  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A001037, A027375, A179043, A184271, A323351.

Cf. A323858, A323859, A323860, A323861, A323864, A323871.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);
Table[If[n==0,1,Length[Union[First/@matcyc/@Select[zaz[n],And[apermatQ[#],neckmatQ[#]]&]]]],{n,0,10}]

a(n) = Sum_{d|n} A323861(d, n/d) for n > 0. - Andrew Howroyd, Aug 21 2019

Terms a(19) and beyond from Andrew Howroyd, Aug 21 2019

A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

Original entry on oeis.org

2, 2, 2, 6, 8, 6, 12, 54, 54, 12, 30, 216, 486, 216, 30, 54, 990, 4020, 4020, 990, 54, 126, 3912, 32730, 64800, 32730, 3912, 126, 240, 16254, 261414, 1047540, 1047540, 261414, 16254, 240, 504, 64800, 2097018, 16764840, 33554250, 16764840, 2097018, 64800, 504

Offset: 1

Gus Wiseman, Feb 04 2019

The 1-dimensional case is A027375.

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			Table begins:
       1     2     3     4
    ------------------------
  1: |  2     2     6    12
  2: |  2     8    54   216
  3: |  6    54   486  4020
  4: | 12   216  4020 64800
The A(2,2) = 8 arrays:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are independently aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

First and last columns are A027375. Main diagonal is A323863.
Cf. A000740, A001037, A179043, A265627, A323351.
Cf. A323861, A323862, A323864, A323865, A323867, A323869.

# See A323861 for code.
for n in [1..8] do for k in [1..8] do Print(n*k*A323861(n,k), ", "); od; Print("\n"); od; # Andrew Howroyd, Aug 21 2019

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],apermatQ]],{n,8},{k,n-1}]

T(n,k) = n*k*A323861(n,k). - Andrew Howroyd, Aug 21 2019

Terms a(29) and beyond from Andrew Howroyd, Aug 21 2019

A323295 Number of ways to fill a matrix with the first n positive integers.

Original entry on oeis.org

1, 1, 4, 12, 72, 240, 2880, 10080, 161280, 1088640, 14515200, 79833600, 2874009600, 12454041600, 348713164800, 5230697472000, 104613949440000, 711374856192000, 38414242234368000, 243290200817664000, 14597412049059840000, 204363768686837760000

Offset: 0

Gus Wiseman, Jan 12 2019

nonn

			The a(4) = 72 matrices consist of:
  24 row/column permutations of [1 2 3 4]
+
  4 row/column permutations of [1 2]
                               [3 4]
+
  4 row/column permutations of [1 2]
                               [4 3]
+
  4 row/column permutations of [1 3]
                               [2 4]
+
  4 row/column permutations of [1 3]
                               [4 2]
+
  4 row/column permutations of [1 4]
                               [2 3]
+
  4 row/column permutations of [1 4]
                               [3 2]
+
  24 row/column permutations of [1]
                                [2]
                                [3]
                                [4]

Cf. A000005, A000142, A038041, A053529, A057625, A061095, A120733, A121860.

Cf. A323300, A323301, A323307, A323351.

Join[{1}, Table[DivisorSigma[0, n]*n!, {n, 30}]]

a(n) = if (n==0, 1, numdiv(n)*n!); \\ Michel Marcus, Jan 15 2019

a(n) = A000005(n) * n! for n > 0, a(0) = 1.

E.g.f.: 1 + Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 13 2019

A323859 Number of binary toroidal necklaces of size n.

Original entry on oeis.org

1, 2, 6, 8, 19, 16, 56, 40, 152, 184, 432, 376, 2132, 1264, 4728, 8768, 20688, 15424, 87656, 55192, 315128, 399520, 762984, 729448, 5595408, 4026576, 10325712, 19884504, 57527804, 37025584, 286340544, 138547336, 805335364, 1041204704, 2021176512, 3926827328

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional (necklace) case is A000031.

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. Alternatively, a toroidal necklace is a matrix that is minimal among all possible rotations of its sequence of rows and its sequence of columns.

			Inequivalent representatives of the a(4) = 19 binary toroidal necklaces:
  [0 0 0 0] [0 0 0 1] [0 0 1 1] [0 1 0 1] [0 1 1 1] [1 1 1 1]
.
  [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]
  [0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]
.
  [0] [0] [0] [0] [0] [1]
  [0] [0] [0] [1] [1] [1]
  [0] [0] [1] [0] [1] [1]
  [0] [1] [1] [1] [1] [1]

Andrew Howroyd, Table of n, a(n) for n = 0..1000
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A008965, A179043, A184271, A323351.

Cf. A323858, A323859, A323865, A323870.

matcyc[m_]:=Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[If[n==0,1,Length[Union[First/@matcyc/@Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n])]]],{n,0,10}]

U(n, m, k) = (1/(n*m)) * sumdiv(n, c, sumdiv(m, d, eulerphi(c) * eulerphi(d) * k^(n*m/lcm(c, d))))
a(n) = if(n<1, n==0, sumdiv(n, d, U(n/d, d, 2))) \\ Andrew Howroyd, Jan 24 2023

a(n) = (1/n) * Sum_{d|n} Sum_{e|d, f|(n/d)} phi(e) * phi(f) * 2^(n/lcm(d,n/d)). [Ethier]

A323307 Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 3, 12, 18, 12, 2, 36, 4, 10, 20, 72, 2, 60, 4, 40, 60, 24, 3, 120, 80, 14, 360, 120, 4, 240, 2, 240, 42, 32, 70, 720, 6, 27, 112, 480, 2, 210, 4, 84, 420, 40, 4, 1440, 280, 280, 108, 224, 5, 1260, 224, 420, 180, 22, 2, 840, 6, 72, 1680, 2880

Offset: 1

Gus Wiseman, Jan 13 2019

nonn

This multiset (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}.

			The a(22) = 24 matrices:
  [111112] [111121] [111211] [112111] [121111] [211111]
.
  [111] [111] [111] [112] [121] [211]
  [112] [121] [211] [111] [111] [111]
.
  [11] [11] [11] [11] [12] [21]
  [11] [11] [12] [21] [11] [11]
  [12] [21] [11] [11] [11] [11]
.
  [1] [1] [1] [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] [1]
  [2] [1] [1] [1] [1] [1]

Gus Wiseman, The a(27) = 360 matrix-arrangements of (1,1,2,2,3,3).

Cf. A007716, A056239, A112798, A120733, A181821, A305936, A318283, A318284, A323295, A323300, A323351.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Array[Length[ptnmats[Times@@Prime/@nrmptn[#]]]&,30]

a(n) = A318762(n) * A000005(A056239(n)).

A323301 Number of ways to fill a matrix with the parts of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 21, 25, 37, 53, 137, 153, 249, 337, 505, 845, 1085, 1497, 2061, 2785, 3661, 7589, 8849, 13329, 18033, 26017, 34225, 48773, 70805, 91977, 123765, 164761, 216373, 283205, 367913, 470889, 758793, 913825, 1264105, 1651613, 2251709, 2894793, 3927837

Offset: 0

Gus Wiseman, Jan 12 2019

nonn

			The a(6) = 21 matrices:
  [6] [1 5] [5 1] [2 4] [4 2] [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
.
  [1] [5] [2] [4]
  [5] [1] [4] [2]
.
  [1] [1] [2] [2] [3] [3]
  [2] [3] [1] [3] [1] [2]
  [3] [2] [3] [1] [2] [1]

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A000005, A000009, A000142, A053529, A294617, A321654, A323295, A323300, A323307, A323351.

b:= proc(n, i, t) option remember;
      `if`(n>i*(i+1)/2, 0, `if`(n=0, t!*numtheory[tau](t),
       b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Sum[Length[ptnmats[k]],{k,Select[Times@@Prime/@#&/@IntegerPartitions[n],SquareFreeQ]}],{n,20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0,
     If[n == 0, t!*DivisorSigma[0, t], b[n, i - 1, t] +
     b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

a(n) = Sum_{y1 + ... + yk = n, y1 > ... > yk} k! * A000005(k) for n > 0, a(0) = 1.

a(0)=1 prepended by Alois P. Heinz, Jan 15 2019

A323864 Number of aperiodic binary arrays of size n.

Original entry on oeis.org

1, 2, 4, 12, 32, 60, 216, 252, 912, 1494, 3960, 4092, 23904, 16380, 65016, 130920, 324960, 262140, 1569132, 1048572, 6281280, 8388072, 16769016, 16777212, 134150880, 100663050, 268402680, 536865840, 1610449344, 1073741820, 8589664080, 4294967292, 25768888320

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(4) = 32 arrays:
  [0001][0010][0011][0100][0110][0111][1000][1001][1011][1100][1101][1110]
.
  [00] [00] [01] [01] [10] [10] [11] [11]
  [01] [10] [00] [11] [00] [11] [01] [10]
.
  [0] [0] [0] [0] [0] [0] [1] [1] [1] [1] [1] [1]
  [0] [0] [0] [1] [1] [1] [0] [0] [0] [1] [1] [1]
  [0] [1] [1] [0] [1] [1] [0] [0] [1] [0] [0] [1]
  [1] [0] [1] [0] [0] [1] [0] [1] [1] [0] [1] [0]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000740, A027375, A265627, A323351.

Cf. A323860, A323862, A323863, A323865, A323867, A323869.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);
Table[Length[Select[zaz[n],apermatQ]],{n,10}]

a(n) = Sum_{d|n} A323860(d, n/d). - Andrew Howroyd, Aug 21 2019

Terms a(18) and beyond from Andrew Howroyd, Aug 21 2019

A323863 Number of n X n aperiodic binary arrays.

Original entry on oeis.org

1, 2, 8, 486, 64800, 33554250, 68718675672, 562949953420302, 18446744060824780800, 2417851639229257812542976, 1267650600228226023797043513000, 2658455991569831745807614120560664598, 22300745198530623141521551172073990303938400

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(2) = 8 arrays are:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are (independently) aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000031, A000740, A027375, A179043, A265627, A323351.

Cf. A323860, A323862, A323864, A323865, A323867, A323869.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],apermatQ]],{n,4}]

a(n) = 2^(n^2) - (n+1)*2^n + 2*n if n is prime. - Robert Israel, Feb 04 2019

a(n) = n^2 * A323872(n). - Andrew Howroyd, Aug 21 2019

a(5) from Robert Israel, Feb 04 2019
a(6)-a(7) from Giovanni Resta, Feb 05 2019
Terms a(8) and beyond from Andrew Howroyd, Aug 21 2019

A323862 Table read by antidiagonals where A(n,k) is the number of n X k binary arrays in which both the sequence of rows and the sequence of columns are (independently) aperiodic.

Original entry on oeis.org

2, 2, 2, 6, 10, 6, 12, 54, 54, 12, 30, 228, 498, 228, 30, 54, 990, 4020, 4020, 990, 54, 126, 3966, 32730, 65040, 32730, 3966, 126, 240, 16254, 261522, 1047540, 1047540, 261522, 16254, 240, 504, 65040, 2097018, 16768860, 33554370, 16768860, 2097018, 65040, 504

Offset: 1

Gus Wiseman, Feb 04 2019

A sequence of length n is aperiodic if all n rotations of its entries are distinct.

			Array begins:
        2        2        6       12       30
        2       10       54      228      990
        6       54      498     4020    32730
       12      228     4020    65040  1047540
       30      990    32730  1047540 33554370

Andrew Howroyd, Table of n, a(n) for n = 1..1275

First and last columns are A027375. Main diagonal is A265627.
Cf. A000031, A000740, A001037, A179043, A323351.
Cf. A323860, A323861, A323863, A323864, A323865, A323867, A323869.

nn=5;
a[n_,k_]:=Sum[MoebiusMu[d]*MoebiusMu[e]*2^(n/d*k/e),{d,Divisors[n]},{e,Divisors[k]}];
Table[a[n-k,k],{n,nn},{k,n-1}]

A(n,k) = {sumdiv(n, d, sumdiv(k,e, moebius(d) * moebius(e) * 2^((n/d) * (k/e))))} \\ Andrew Howroyd, Jan 19 2023

A(n,k) = Sum_{d|n, e|k} mu(d) * mu(e) * 2^((n/d) * (k/e)).

cp's OEIS Frontend

A323300 Number of ways to fill a matrix with the parts of the integer partition with Heinz number n.

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A323865 Number of aperiodic binary toroidal necklaces of size n.

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A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

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A323295 Number of ways to fill a matrix with the first n positive integers.

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A323859 Number of binary toroidal necklaces of size n.

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A323307 Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.

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A323301 Number of ways to fill a matrix with the parts of a strict integer partition of n.

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