A340930 Heinz numbers of integer partitions of even negative rank.

8, 24, 32, 36, 54, 80, 81, 96, 120, 128, 144, 180, 200, 216, 224, 270, 300, 320, 324, 336, 384, 405, 450, 480, 486, 500, 504, 512, 560, 576, 675, 704, 720, 729, 750, 756, 784, 800, 840, 864, 896, 1056, 1080, 1125, 1134, 1176, 1200, 1250, 1260, 1280, 1296, 1344

Offset: 1

Author

Gus Wiseman, Jan 30 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:
       8: (1,1,1)             270: (3,2,2,2,1)
      24: (2,1,1,1)           300: (3,3,2,1,1)
      32: (1,1,1,1,1)         320: (3,1,1,1,1,1,1)
      36: (2,2,1,1)           324: (2,2,2,2,1,1)
      54: (2,2,2,1)           336: (4,2,1,1,1,1)
      80: (3,1,1,1,1)         384: (2,1,1,1,1,1,1,1)
      81: (2,2,2,2)           405: (3,2,2,2,2)
      96: (2,1,1,1,1,1)       450: (3,3,2,2,1)
     120: (3,2,1,1,1)         480: (3,2,1,1,1,1,1)
     128: (1,1,1,1,1,1,1)     486: (2,2,2,2,2,1)
     144: (2,2,1,1,1,1)       500: (3,3,3,1,1)
     180: (3,2,2,1,1)         504: (4,2,2,1,1,1)
     200: (3,3,1,1,1)         512: (1,1,1,1,1,1,1,1,1)
     216: (2,2,2,1,1,1)       560: (4,3,1,1,1,1)
     224: (4,1,1,1,1,1)       576: (2,2,1,1,1,1,1,1)
		

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 A101708.

The positive version is (A340605).

The odd version is A101707 (A340929).

The not necessarily even version is A064173 (A340788).

A001222 counts prime factors.

A027187 counts partitions of even length.

A047993 counts balanced partitions (A106529).

A056239 adds up prime indices.

A058696 counts partitions of even numbers.

A061395 selects the maximum prime index.

A063995/A105806 count partitions by Dyson rank.

A072233 counts partitions by sum and length.

A112798 lists the prime indices of each positive integer.

A168659 counts partitions whose length is divisible by maximum.

A200750 counts partitions whose length and maximum are relatively prime.

- 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.

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

A340601 counts partitions of even rank (A340602).

A340692 counts partitions of odd rank (A340603).

Cf. A003114, A006141, A039900, A117193, A117409, A316413, A324517, A325134, A326845, A340604, A340787, A340828, A340830.

Programs

Mathematica

rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n];
Select[Range[2,100],EvenQ[rk[#]]&&rk[#]<0&]