A114640 -id:A114640 - cp's OEIS frontend

A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

1, 1, 0, 1, 3, 2, 3, 2, 5, 5, 10, 9, 14, 14, 21, 20, 30, 36, 44, 50, 66, 75, 93, 106, 132, 151, 185, 212, 256, 286, 348, 394, 479, 543, 642, 740, 888, 994, 1176, 1350, 1589, 1789, 2109, 2371, 2786, 3144, 3653, 4126, 4811, 5385, 6213

Offset: 0

Gus Wiseman, Nov 30 2023

nonn

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

			The a(1) = 1 through a(10) = 10 partitions:
  (1)  .  (21)  (22)   (41)   (51)    (61)   (71)     (81)    (91)
                (31)   (221)  (321)   (421)  (431)    (333)   (541)
                (211)         (3111)         (521)    (531)   (631)
                                             (3221)   (621)   (721)
                                             (41111)  (4221)  (3322)
                                                              (3331)
                                                              (4321)
                                                              (5221)
                                                              (322111)
                                                              (511111)

The case of strict partitions is A025147, ranks A039956.
The case of distinct multiplicities is A114640, ranks A109297.
These partitions have ranks A367683.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
Cf. A000837, A071625, A072774, A082090, A116861, A367579, A367580, A367581.

mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], Sort[Length/@Split[#]]==mmk[#]&]], {n,0,15}]

A367684 Number of integer partitions of n whose multiset multiplicity kernel is a submultiset.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 10, 14, 17, 25, 30, 39, 51, 66, 79, 102, 125, 154, 191, 233, 284, 347, 420, 499, 614, 726, 867, 1031, 1233, 1437, 1726, 2002, 2375, 2770, 3271, 3760, 4455, 5123, 5994, 6904, 8064, 9199, 10753, 12241, 14202, 16189, 18704, 21194, 24504

Offset: 0

Gus Wiseman, Nov 30 2023

nonn

We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.

			The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (2211)    (2221)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)

The case of strict partitions is A000012.
Includes all partitions with distinct multiplicities A098859, ranks A130091.
These partitions have ranks A367685.
A000041 counts integer partitions, strict A000009.
A072233 counts partitions by number of parts.
A091602 counts partitions by greatest multiplicity, least A243978.
A116608 counts partitions by number of distinct parts.
A116861 counts partitions by sum of distinct parts.
Cf. A000837, A032741, A071625, A109297, A114640, A367579, A367580, A367581.

submultQ[cap_,fat_]:=And@@Function[i, Count[fat,i]>=Count[cap, i]]/@Union[List@@cap];
mmk[q_List]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
Table[Length[Select[IntegerPartitions[n], submultQ[mmk[#],#]&]], {n,0,15}]

A336031 Number of compositions of n such that the set of parts and the set of multiplicities of parts are equal.

Original entry on oeis.org

1, 1, 0, 0, 4, 3, 4, 0, 5, 1, 70, 120, 122, 130, 446, 0, 277, 726, 370, 1064, 13751, 38913, 41272, 81168, 137014, 84448, 300642, 490540, 517806, 341033, 1467180, 425328, 2403512, 2916863, 4455856, 39855808, 164203236, 216675811, 447273890, 730795760, 1154455982

Offset: 0

Alois P. Heinz, Jul 07 2020

nonn

Cf. A114640, A336032.

b:= proc(n, i, p, f, g) option remember; `if`(n=0, `if`(f=g, p!, 0),
      `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0, f, {f[], i}),
      `if`(j=0, g, {g[], j}))/j!, j=0..n/i)))
    end:
a:= n-> b(n$2, 0, {}$2):
seq(a(n), n=0..32);

b[n_, i_, p_, f_, g_] := b[n, i, p, f, g] = If[n == 0, If[f == g, p!, 0],
     If[i < 1, 0, Sum[b[n - i*j, i - 1, p + j,
     If[j == 0, f, Union@Append[f, i]],
     If[j == 0, g, Union@Append[g, j]]]/j!, {j, 0, n/i}]]];
a[n_] := b[n, n, 0, {}, {}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 13 2022, after Alois P. Heinz *)

A353504 Number of integer partitions of n whose product is less than the product of their multiplicities.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 4, 4, 6, 6, 8, 9, 11, 12, 14, 14, 18, 21, 23, 26, 29, 29, 33, 36, 39, 40, 43, 44, 50, 53, 55, 59, 65, 69, 72, 78, 79, 81, 85, 92, 95, 97, 100, 103, 108, 109, 112, 118, 124, 129, 137, 139, 142, 149, 155, 159, 165, 166, 173, 178, 181, 187

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(2) = 1 through a(9) = 6 partitions:
  11   111   1111   2111    21111    22111     221111     222111
                    11111   111111   31111     311111     411111
                                     211111    2111111    2211111
                                     1111111   11111111   3111111
                                                          21111111
                                                          111111111

LHS (product of parts) is counted by A339095, ranked by A003963.
RHS (product of multiplicities) is counted by A266477, ranked by A005361.
The version for greater instead of less is A353505.
The version for equal instead of less is A353506, ranked by A353503.
A181819 gives prime shadow, with an inverse A181821.
A353398 counts partitions with the same product of multiplicities as of shadows, ranked by A353399.
Cf. A002033, A008284, A085629, A097318, A098859, A114640, A116608, A118914, A124010, A304678, A353394, A353500, A353507.

Table[Length[Select[IntegerPartitions[n],Times@@#

A353505 Number of integer partitions of n whose product is greater than the product of their multiplicities.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 11, 17, 24, 35, 47, 66, 89, 121, 162, 214, 276, 362, 464, 599, 763, 971, 1219, 1537, 1918, 2393, 2966, 3668, 4512, 5549, 6784, 8287, 10076, 12238, 14807, 17898, 21556, 25931, 31094, 37243, 44486, 53075, 63158, 75069, 89025, 105447, 124636

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(0) = 0 through a(7) = 11 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)    (7)
             (21)  (22)  (32)   (33)   (43)
                   (31)  (41)   (42)   (52)
                         (221)  (51)   (61)
                         (311)  (222)  (322)
                                (321)  (331)
                                (411)  (421)
                                       (511)
                                       (2221)
                                       (3211)
                                       (4111)

RHS (product of multiplicities) is counted by A266477, ranked by A005361.
LHS (product of parts) is counted by A339095, ranked by A003963.
The version for less instead of greater is A353504.
The version for equality is A353506, ranked by A353503.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353398 counts partitions with the same products of multiplicities as of shadows, ranked by A353399.
Cf. A002033, A008284, A008619, A085629, A097318, A098859, A114640, A116608, A130091, A304678, A353394, A353507.

Table[Length[Select[IntegerPartitions[n],Times@@#>Times@@Length/@Split[#]&]],{n,0,30}]

A353696 Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence that is already counted.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 16, 32, 43, 58, 64, 128, 256, 292, 349, 442, 512, 586, 676, 697, 826, 1024, 1210, 1338, 1393, 1394, 1396, 1594, 2048, 2186, 2234, 2618, 2696, 2785, 2786, 2792, 3130, 4096, 4282, 4410, 4666, 5178, 5569, 5570, 5572, 5576, 5584, 6202, 8192

Offset: 1

Gus Wiseman, May 22 2022

nonn

First differs from the non-consecutive version A353431 in lacking 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are a subsequence but not a consecutive subsequence.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
    0: ()
    1: (1)
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   32: (6)
   43: (2,2,1,1)
   58: (1,1,2,2)
   64: (7)
  128: (8)
  256: (9)
  292: (3,3,3)
  349: (2,2,1,1,2,1)
  442: (1,2,1,1,2,2)
  512: (10)
  586: (3,3,2,2)
  676: (2,2,3,3)
  697: (2,2,1,1,3,1)
  826: (1,3,1,1,2,2)

Non-recursive non-consecutive for partitions: A325755, counted by A325702.
Non-consecutive: A353431, counted by A353391.
Non-consecutive for partitions: A353393, counted by A353426.
Non-recursive non-consecutive: A353402, counted by A353390.
Counted by: A353430.
Non-recursive: A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, run-lengths A333769.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, contiguous A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, multisets A225620, sets A333256.
- Runs are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Anti-runs are A333489, counted by A003242.
Cf. A032020, A114640, A181819, A228351, A329739, A318928, A325705, A329738, A333224, A353427, A353403.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]]&&yoyQ[Length/@Split[y]];
Select[Range[0,1000],yoyQ[stc[#]]&]

A325766 Number of integer partitions of n covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 4, 5, 4, 6, 7, 8, 6, 12, 11, 19, 16, 22, 22, 25, 32, 38, 45, 45, 51, 53, 71, 69, 85, 92, 118, 125, 147, 149, 184, 187, 230, 254, 290, 317, 372, 397, 449, 502, 544, 616, 680, 758, 841, 930, 1042, 1151, 1262

Offset: 0

Gus Wiseman, May 19 2019

nonn

The Heinz numbers of these partitions are given by A325767.

			The initial terms count the following partitions:
   1: (1)
   4: (2,1,1)
   5: (2,2,1)
   6: (2,2,1,1)
   7: (3,2,1,1)
   8: (3,2,1,1,1)
   9: (3,2,2,1,1)
  10: (3,2,2,1,1,1)
  11: (3,3,2,2,1)
  11: (3,3,2,1,1,1)
  11: (3,2,2,2,1,1)
  12: (4,3,2,1,1,1)
  13: (4,3,2,2,1,1)
  13: (4,3,2,1,1,1,1)
  13: (3,3,3,2,1,1)
  13: (3,3,2,2,2,1)
  13: (3,3,2,2,1,1,1)
  14: (4,3,2,2,1,1,1)
  14: (3,3,3,2,2,1)
  14: (3,3,3,2,1,1,1)
  14: (3,3,2,2,2,1,1)

Cf. A000009 (partitions covering an initial interval), A055932, A114639, A114640, A290689, A324753, A325702, A325706, A325707, A325708, A325767.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap]
Table[Length[Select[IntegerPartitions[n],Range[Length[Union[#]]]==Union[#]&&submultQ[Sort[Length/@Split[#]],Sort[#]]&]],{n,0,30}]

A325767 Heinz numbers of integer partitions covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

Original entry on oeis.org

1, 2, 12, 18, 36, 60, 120, 180, 360, 450, 540, 600, 840, 1260, 1350, 1500, 1680, 1800, 2250, 2520, 2700, 3000, 3780, 4200, 4500, 5040, 5400, 5880, 6750, 8400, 9000, 10500, 11340, 11760, 12600, 13500, 15120, 17640, 18480, 18900, 20580, 21000, 22680, 25200

Offset: 1

Gus Wiseman, May 19 2019

nonn

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

The enumeration of these partitions by sum is given by A325766.

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
    12: {1,1,2}
    18: {1,2,2}
    36: {1,1,2,2}
    60: {1,1,2,3}
   120: {1,1,1,2,3}
   180: {1,1,2,2,3}
   360: {1,1,1,2,2,3}
   450: {1,2,2,3,3}
   540: {1,1,2,2,2,3}
   600: {1,1,1,2,3,3}
   840: {1,1,1,2,3,4}
  1260: {1,1,2,2,3,4}
  1350: {1,2,2,2,3,3}
  1500: {1,1,2,3,3,3}
  1680: {1,1,1,1,2,3,4}
  1800: {1,1,1,2,2,3,3}
  2250: {1,2,2,3,3,3}
  2520: {1,1,1,2,2,3,4}

Cf. A000009, A055932, A056239, A112798, A109297, A114640, A290689, A325702, A325708, A325766.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[1000],#==1||Range[PrimeNu[#]]==PrimePi/@First/@FactorInteger[#]&&Divisible[#,red[#]]&]

A353699 Heinz numbers of integer partitions whose product equals their length.

Original entry on oeis.org

2, 6, 20, 36, 56, 176, 240, 416, 864, 1088, 1344, 2432, 3200, 5888, 8448, 14848, 23040, 31744, 35840, 39936, 75776, 167936, 208896, 331776, 352256, 450560, 516096, 770048, 802816, 933888, 1736704, 2457600, 3866624, 4259840, 4521984, 7995392, 12976128, 17563648

Offset: 1

Gus Wiseman, May 19 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      2: {1}
      6: {1,2}
     20: {1,1,3}
     36: {1,1,2,2}
     56: {1,1,1,4}
    176: {1,1,1,1,5}
    240: {1,1,1,1,2,3}
    416: {1,1,1,1,1,6}
    864: {1,1,1,1,1,2,2,2}
   1088: {1,1,1,1,1,1,7}
   1344: {1,1,1,1,1,1,2,4}
   2432: {1,1,1,1,1,1,1,8}
   3200: {1,1,1,1,1,1,1,3,3}
   5888: {1,1,1,1,1,1,1,1,9}
   8448: {1,1,1,1,1,1,1,1,2,5}
  14848: {1,1,1,1,1,1,1,1,1,10}
  23040: {1,1,1,1,1,1,1,1,1,2,2,3}
  31744: {1,1,1,1,1,1,1,1,1,1,11}
  35840: {1,1,1,1,1,1,1,1,1,1,3,4}
  39936: {1,1,1,1,1,1,1,1,1,1,2,6}
  75776: {1,1,1,1,1,1,1,1,1,1,1,12}

Length is A001222, counted by A008284, distinct A001221.
Product is A003963, counted by A339095, firsts A318871.
A similar sequence is A353503, counted by A353506.
These partitions are counted by A353698.
A005361 gives product of signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353394 gives product of shadows of prime indices, firsts A353397.
Cf. A000720, A003586, A114640, A182850, A320325, A325131, A325755, A353399, A353507.

Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==PrimeOmega[#]&]

cp's OEIS Frontend

A367682 Number of integer partitions of n whose multiset of multiplicities is the same as their multiset multiplicity kernel.

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A367684 Number of integer partitions of n whose multiset multiplicity kernel is a submultiset.

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A336031 Number of compositions of n such that the set of parts and the set of multiplicities of parts are equal.

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A353504 Number of integer partitions of n whose product is less than the product of their multiplicities.

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A353505 Number of integer partitions of n whose product is greater than the product of their multiplicities.

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A353696 Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence that is already counted.

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A325766 Number of integer partitions of n covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

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A325767 Heinz numbers of integer partitions covering an initial interval of positive integers and containing their own multiset of multiplicities (as a submultiset).

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Author

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Mathematica

A353699 Heinz numbers of integer partitions whose product equals their length.

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Mathematica