A340788 Heinz numbers of integer partitions of negative rank.

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 144, 150, 160, 162, 168, 180, 192, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 320, 324, 336, 352, 360, 375, 378, 384, 392, 400, 405

Offset: 1

Author

Gus Wiseman, Jan 29 2021

Keywords

• nonn

Comments

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

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

Examples

			The sequence of partitions together with their Heinz numbers begins:
      4: (1,1)             80: (3,1,1,1,1)
      8: (1,1,1)           81: (2,2,2,2)
     12: (2,1,1)           90: (3,2,2,1)
     16: (1,1,1,1)         96: (2,1,1,1,1,1)
     18: (2,2,1)          100: (3,3,1,1)
     24: (2,1,1,1)        108: (2,2,2,1,1)
     27: (2,2,2)          112: (4,1,1,1,1)
     32: (1,1,1,1,1)      120: (3,2,1,1,1)
     36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
     40: (3,1,1,1)        135: (3,2,2,2)
     48: (2,1,1,1,1)      144: (2,2,1,1,1,1)
     54: (2,2,2,1)        150: (3,3,2,1)
     60: (3,2,1,1)        160: (3,1,1,1,1,1)
     64: (1,1,1,1,1,1)    162: (2,2,2,2,1)
     72: (2,2,1,1,1)      168: (4,2,1,1,1)
		

Links

• Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
• FindStat, St000145: The Dyson rank of a partition

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.

These partitions are counted by A064173.

The odd case is A101707 is (A340929).

The even case is A101708 is (A340930).

The positive version is (A340787).

A001222 counts prime factors.

A061395 selects the maximum prime index.

A072233 counts partitions by sum and length.

A168659 counts partitions whose length is divisible by maximum.

A200750 counts partitions whose length and maximum are relatively prime.

- Rank -

A047993 counts partitions of rank 0 (A106529).

A063995/A105806 count partitions by Dyson rank.

A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).

A101198 counts partitions of rank 1 (A325233).

A257541 gives the rank of the partition with Heinz number n.

A324518 counts partitions with rank equal to greatest part (A324517).

A324520 counts partitions with rank equal to least part (A324519).

A340601 counts partitions of even rank (A340602), with strict case A117192.

A340692 counts partitions of odd rank (A340603), with strict case A117193.

Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A325134, A326845, A340604, A340605, A340828.

Programs

Mathematica

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]
				

Formula

For all terms A061395(a(n)) < A001222(a(n)).