A325702 -id:A325702 - cp's OEIS frontend

A325707 Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.

1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 4, 4, 5, 6, 7, 8, 10, 11, 13, 16, 18, 23, 26, 32, 36, 43, 48, 57, 64, 74, 84, 98, 110, 127, 145, 165, 189, 215, 244, 277, 316, 356, 403, 455, 513, 577, 650, 727, 817, 913, 1024, 1143, 1279, 1425, 1592, 1773, 1977, 2198, 2448, 2717

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325708.

			The initial terms count the following partitions:
   1: (1)
   3: (21)
   4: (211)
   5: (221)
   6: (321)
   6: (2211)
   7: (3211)
   8: (3221)
   8: (32111)
   9: (3321)
   9: (32211)
  10: (4321)
  10: (33211)
  10: (32221)
  10: (322111)
  11: (43211)
  11: (33221)
  11: (332111)
  11: (322211)
  12: (43221)
  12: (432111)
  12: (33321)
  12: (332211)
  12: (3222111)

Cf. A000009 (partitions covering an initial interval), A055932, A109297, A114639, A114640, A290689, A324753, A325702, A325706, A325708.

Table[Length[Select[IntegerPartitions[n],Range[Length[Union[#]]]==Union[#]&&SubsetQ[Sort[#],Sort[Length/@Split[#]]]&]],{n,0,30}]

A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 3, 1, 3, 4, 3, 7, 5, 9, 8, 12, 15, 15, 20, 21, 25, 31, 33, 38, 42, 46, 56, 61, 67, 78, 76, 96, 100, 114, 131, 130, 157, 157, 185, 200, 214, 236, 253, 275, 302, 333, 351, 386, 408, 440, 486, 515, 564, 596, 633, 691, 734, 800, 854, 899, 964

Offset: 0

Gus Wiseman, May 15 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.

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The a(8) = 1 through a(14) = 9 partitions (A..D = 10..13):
  (53)  (72)    (73)    (B)     (75)     (D)      (B3)
        (621)   (532)   (A1)    (651)    (B2)     (752)
        (4221)  (631)   (4331)  (732)    (A21)    (761)
                (4411)          (6321)   (43321)  (A31)
                                (6411)   (44311)  (C11)
                                (43221)           (6521)
                                (44211)           (9221)
                                                  (54221)
                                                  (64211)

The LHS (prime shadow) is A181819, with an inverse A181821.
The RHS (product of prime shadows) is A353394, first appearances A353397.
These partitions are ranked by A353395.
A related comparison is A353398, ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A239455 counts Look-and-Say partitions, ranked by A351294.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000005, A002033, A143773, A182850, A316428, A325131, A325702, A325755, A353389, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==red[Times@@Prime/@#]&]],{n,0,15}]

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[#]]&]

cp's OEIS Frontend

A325707 Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.

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A353396 Number of integer partitions of n whose Heinz number has prime shadow equal to the product of prime shadows of its parts.

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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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Mathematica

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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Mathematica