A225486 -id:A225486 - cp's OEIS frontend

A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

0, 1, 1, 1, 1, 3, 3, 1, 3, 7, 10, 17, 27, 38, 1, 4, 8, 17, 31, 52, 83, 122, 181, 257, 361, 499, 684, 910, 1211, 1595, 2060, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 1, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754, 2570, 3742, 5269

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

The Heinz numbers of these partitions are given by A325283.
The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014. The maximum adjusted frequency depth for integer partitions of n is given by A325282.
Essentially, the last numbers of rows of the array in A225485. - Clark Kimberling, Sep 13 2022

			The a(1) = 1 through a(11) = 17 partitions:
  1  11  21  211  221   411    3211  3221   3321    5221     4322
                  311   3111         4211   4221    5311     4331
                  2111  21111        32111  4311    6211     4421
                                            5211    32221    5411
                                            32211   33211    6221
                                            42111   42211    6311
                                            321111  43111    7211
                                                    52111    33221
                                                    421111   42221
                                                    3211111  43211
                                                             52211
                                                             53111
                                                             62111
                                                             431111
                                                             521111
                                                             4211111
                                                             32111111

Cf. A011784, A181819, A182850, A182857, A225486, A323014, A323023, A325246, A325258, A325278, A325281, A325282, A325283.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

nn=30;
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==mfds[[n]]&]],{n,0,nn}]

A377056 Antidiagonal-sums of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 3, 11, 2, 36, -27, 142, -207, 595, -1066, 2497, -4878, 10726, -22189, 48383, -103318, 224296, -480761, 1030299, -2186942, 4626313, -9740648, 20492711, -43109372, 90843475, -191769296, 405528200, -858373221, 1817311451, -3845483855, 8129033837

Offset: 0

Gus Wiseman, Dec 12 2024

sign

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 3.

For primes we have A140119 or A376683, unsigned A376681 or A376684.
These are the antidiagonal-sums of A175804.
First column of the same array is A281425.
For composites we have A377034, unsigned A377035.
For squarefree numbers we have A377039, unsigned A377040.
For nonsquarefree numbers we have A377049, unsigned A377048.
For prime powers we have A377052, unsigned A377053.
The unsigned version is A378621.
The version for strict partitions is A378970 (row-sums of A378622), unsigned A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A116608, A225486, A293467, A325242, A325245, A325268, A377285.

nn=20;
t=Table[Differences[PartitionsP/@Range[0,2nn],k],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn},{j,i}]

A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

Original entry on oeis.org

0, 1, 1, 5, 5, 8, 20, 7, 22

Offset: 0

Gus Wiseman, Dec 12 2024

Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.

a(12) = 47. Conjecture: a(n) = 0 for n > 12. - Chai Wah Wu, Dec 15 2024

			The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.

For primes we have A376678.
For composites we have A377037.
For squarefree numbers we have A377042.
For nonsquarefree numbers we have A377050.
For prime-powers we have A377055.
Position of first zero in each row of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives first column (up to sign).
- A378970 gives row-sums.
- A378971 gives row-sums of absolute value.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A047966, A098859, A225486, A325244, A325282.
Cf. A008284, A116608, A325242, A325268, A225485 or A325280.

Table[Position[Differences[PartitionsQ/@Range[0,100],k],0][[1,1]],{k,1,8}]

a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w;); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024

A378970 Antidiagonal-sums of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, -4, 18, -20, 47, -56, 110, -153, 309, -532, 1045, -1768, 2855, -3620, 2928, 2927, -20371, 62261, -148774, 314112, -613835, 1155936, -2175658, 4244218, -8753316, 19006746, -42471491, 95234915, -210395017, 453414314, -949507878, 1931940045

Offset: 0

Gus Wiseman, Dec 14 2024

sign

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = -4.

For primes we have A140119 or A376683, absolute value A376681 or A376684.
For composites we have A377034, absolute value A377035.
For squarefree numbers we have A377039, absolute value A377040.
For nonsquarefree numbers we have A377047, absolute value A377048.
For prime powers we have A377052, absolute value A377053.
For partition numbers we have A377056, absolute value A378621.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The unsigned version is A378971.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

Original entry on oeis.org

1, 1, 1, 5, 8, 18, 30, 47, 70, 110, 177, 309, 574, 1063, 1892, 3107, 4598, 6166, 8737, 20603, 62457, 149132, 314116, 614093, 1155968, 2176048, 4244322, 8753864, 19006756, 42472117, 95235017, 210396059, 453414950, 949510166, 1931941261, 3826650257, 7400745917

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal 4 of A378622 is (2, 0, -1, -2, -3), so a(4) = 8.

For primes we have A376681 or A376684, signed version A140119 or A376683.
For composites we have A377035, signed version A377034.
For squarefree numbers we have A377040, signed version A377039.
For nonsquarefree numbers we have A377048, signed version A377049.
For prime powers we have A377053, signed version A377052.
For partition numbers we have A378621, signed version A377056.
Row-sums of the triangular form of A378622. See also:
- A175804 is the version for partitions.
- A293467 gives the first column (up to sign).
- A377285 gives position of first zero in each row.
The signed version is A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A098859, A116608, A225486, A325242, A325244, A325245, A325268.

nn=30;
t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A325706 Heinz numbers of integer partitions containing all of their distinct multiplicities.

Original entry on oeis.org

1, 2, 6, 9, 10, 12, 14, 18, 22, 26, 30, 34, 36, 38, 40, 42, 46, 58, 60, 62, 66, 70, 74, 78, 82, 84, 86, 90, 94, 102, 106, 110, 112, 114, 118, 120, 122, 125, 126, 130, 132, 134, 138, 142, 146, 150, 154, 156, 158, 166, 170, 174, 178, 180, 182, 186, 190, 194, 198

Offset: 1

Gus Wiseman, May 18 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers n divisible by the squarefree kernel of their "shadow" A181819(n).
The enumeration of these partitions by sum is given by A325705.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   34: {1,7}
   36: {1,1,2,2}
   38: {1,8}
   40: {1,1,1,3}
   42: {1,2,4}
   46: {1,9}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}

Cf. A056239, A109297, A112798, A114639, A114640, A181819, A225486, A290689, A324753, A324843, A325702, A325705, A325707, A325755.

Select[Range[100],#==1||SubsetQ[PrimePi/@First/@FactorInteger[#],Last/@FactorInteger[#]]&]

A325283 Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.

Original entry on oeis.org

2, 4, 6, 12, 18, 20, 24, 28, 40, 48, 60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280

Offset: 1

Gus Wiseman, Apr 17 2019

The enumeration of these partitions by sum is given by A325254.
The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices and their omega-sequences (see A323023) begins:
  2:   {1}         (1)
  4:   {1,1}       (2,1)
  6:   {1,2}       (2,2,1)
  12:  {1,1,2}     (3,2,2,1)
  18:  {1,2,2}     (3,2,2,1)
  20:  {1,1,3}     (3,2,2,1)
  24:  {1,1,1,2}   (4,2,2,1)
  28:  {1,1,4}     (3,2,2,1)
  40:  {1,1,1,3}   (4,2,2,1)
  48:  {1,1,1,1,2} (5,2,2,1)
  60:  {1,1,2,3}   (4,3,2,2,1)
  84:  {1,1,2,4}   (4,3,2,2,1)
  90:  {1,2,2,3}   (4,3,2,2,1)
  120: {1,1,1,2,3} (5,3,2,2,1)
  126: {1,2,2,4}   (4,3,2,2,1)
  132: {1,1,2,5}   (4,3,2,2,1)
  140: {1,1,3,4}   (4,3,2,2,1)
  150: {1,2,3,3}   (4,3,2,2,1)
  156: {1,1,2,6}   (4,3,2,2,1)
  168: {1,1,1,2,4} (5,3,2,2,1)
  180: {1,1,2,2,3} (5,3,2,2,1)

Cf. A011784, A056239, A112798, A118914, A181819, A182857, A225486, A323014, A323023, A325254, A325258, A325277, A325278, A325282.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

nn=30;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
Select[Range[Prime[nn]],fdadj[primeMS[#]]==mfds[[Total[primeMS[#]]]]&]

A325414 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 1, 0, 1, 1, 2, 0, 3, 1, 1, 1, 0, 1, 0, 0, 0, 3, 0, 1, 4, 2, 2, 1, 1, 0, 1, 0, 1, 0, 4, 0, 3, 3, 2, 2, 2, 3, 1, 0, 1, 0, 0, 1, 4, 0, 3, 3, 3, 4, 1, 6, 3, 1, 0, 1

Offset: 0

Gus Wiseman, Apr 24 2019

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1) with sum 13, so (32211) is counted under T(9,13).

			Triangle begins:
  1
  0 1
  0 1 0 1
  0 1 0 0 1 1
  0 1 0 1 0 2 0 0 1
  0 1 0 0 0 2 1 0 2 1
  0 1 0 1 1 2 0 3 1 1 1
  0 1 0 0 0 3 0 1 4 2 2 1 1
  0 1 0 1 0 4 0 3 3 2 2 2 3 1
  0 1 0 0 1 4 0 3 3 3 4 1 6 3 1
  0 1 0 1 0 4 1 6 4 4 1 4 5 8 2 1
Row n = 9 counts the following partitions:
  9  333  54  432  441  3222    22221      411111  3321     32211     321111
          63  531  522  6111    33111              4221     42111
          72  621  711  222111  51111              4311     21111111
          81                    111111111          5211
                                                   2211111
                                                   3111111

Row sums are A000041.
Row lengths are A325413(n) + 1 (because k starts at 0).
Number of nonzero terms in row n is A325415(n).
Cf. A181819, A225486, A323014, A323023, A325248, A325249, A325277, A325412, A325415, A325416.
Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (frequency depth), A325414 (omega-sequence sum).

omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n],Total[omseq[#]]==k&]],{n,0,10},{k,0,Max[Total/@omseq/@IntegerPartitions[n]]}]

A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 5, 11, 16, 36, 65, 142, 285, 595, 1210, 2497, 5134, 10726, 22637, 48383, 104066, 224296, 481985, 1030299, 2188912, 4626313, 9743750, 20492711, 43114180, 90843475, 191776658, 405528200, 858384333, 1817311451, 3845500427, 8129033837, 17162815092

Offset: 0

Gus Wiseman, Dec 14 2024

nonn

			Antidiagonal i + j = 3 of A175804 is (3, 1, 0, -1), so a(3) = 5.

These are the antidiagonal-sums of the absolute value of A175804.
First column of the same array is A281425.
For primes we have A376681 or A376684, signed A140119 or A376683.
For composites we have A377035, signed A377034.
For squarefree numbers we have A377040, signed A377039.
For nonsquarefree numbers we have A377048, signed A377049.
For prime powers we have A377053, signed A377052.
The signed version is A377056.
The corresponding array for strict partitions is A378622, see A293467, A377285, A378971, A378970.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865, A053445.
Cf. A008284, A047966, A075526, A098859, A116608, A225486, A325242, A325244, A325268.

nn=30;
q=Table[PartitionsP[n],{n,0,nn}];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}]
Total/@Abs/@Table[t[[j,i-j+1]],{i,nn/2},{j,i}]

A325260 Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 5, 8, 10, 12, 13, 18, 19, 24, 25, 31, 33, 40, 40, 49, 51, 59, 60, 71, 72, 83, 84, 96, 98, 111, 111, 126, 128, 142, 143, 160, 161, 178, 179, 197, 199, 218, 218, 239, 241, 261, 262, 285, 286, 309, 310, 334, 336, 361, 361, 388, 390, 416, 417, 446

Offset: 0

Gus Wiseman, Apr 16 2019

nonn

The omega-sequence of an integer partition is the sequence of lengths of the multisets obtained by repeatedly taking the multiset of multiplicities until a singleton is reached. For example, the partition (32211) has chain of multisets of multiplicities {1,1,2,2,3} -> {1,2,2} -> {1,2} -> {1,1} -> {2}, so its omega-sequence is (5,3,2,2,1).

The Heinz numbers of these partitions are given by A325251.

			The a(1) = 1 through a(9) = 12 partitions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)     (9)
       (11)  (21)  (22)   (32)   (33)   (43)    (44)    (54)
                   (31)   (41)   (42)   (52)    (53)    (63)
                   (211)  (221)  (51)   (61)    (62)    (72)
                          (311)  (411)  (322)   (71)    (81)
                                        (331)   (332)   (441)
                                        (511)   (422)   (522)
                                        (3211)  (611)   (711)
                                                (3221)  (3321)
                                                (4211)  (4221)
                                                        (4311)
                                                        (5211)

Cf. A055932, A181819, A182850, A225486, A323014, A323023, A325250, A325251, A325262, A325277.

Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
omseq[ptn_List]:=If[ptn=={},{},Length/@NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]];
Table[Length[Select[IntegerPartitions[n],normQ[omseq[#]]&]],{n,0,30}]

a(n) + A325262(n) = A000041(n).
Conjectures from Chai Wah Wu, Jan 13 2021: (Start)
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n > 9.
G.f.: (-x^9 - x^8 - x^7 + x^6 - x^5 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)). (End)

cp's OEIS Frontend

A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

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A377056 Antidiagonal-sums of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

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A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

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A378970 Antidiagonal-sums of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

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A378971 Antidiagonal-sums of absolute value of the array A378622(n,k) = n-th term of k-th differences of strict partition numbers (A000009).

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A325706 Heinz numbers of integer partitions containing all of their distinct multiplicities.

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A325283 Heinz numbers of integer partitions with maximum adjusted frequency depth for partitions of that sum.

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A325414 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with omega-sequence summing to n.

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A378621 Antidiagonal-sums of absolute value of the array A175804(n,k) = n-th term of k-th differences of partition numbers (A000041).

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A325260 Number of integer partitions of n whose omega-sequence covers an initial interval of positive integers.