A340929 Heinz numbers of integer partitions of odd negative rank.

4, 12, 16, 18, 27, 40, 48, 60, 64, 72, 90, 100, 108, 112, 135, 150, 160, 162, 168, 192, 225, 240, 243, 250, 252, 256, 280, 288, 352, 360, 375, 378, 392, 400, 420, 432, 448, 528, 540, 567, 588, 600, 625, 630, 640, 648, 672, 700, 768, 792, 810, 832, 880, 882

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)             150: (3,3,2,1)
      12: (2,1,1)           160: (3,1,1,1,1,1)
      16: (1,1,1,1)         162: (2,2,2,2,1)
      18: (2,2,1)           168: (4,2,1,1,1)
      27: (2,2,2)           192: (2,1,1,1,1,1,1)
      40: (3,1,1,1)         225: (3,3,2,2)
      48: (2,1,1,1,1)       240: (3,2,1,1,1,1)
      60: (3,2,1,1)         243: (2,2,2,2,2)
      64: (1,1,1,1,1,1)     250: (3,3,3,1)
      72: (2,2,1,1,1)       252: (4,2,2,1,1)
      90: (3,2,2,1)         256: (1,1,1,1,1,1,1,1)
     100: (3,3,1,1)         280: (4,3,1,1,1)
     108: (2,2,2,1,1)       288: (2,2,1,1,1,1,1)
     112: (4,1,1,1,1)       352: (5,1,1,1,1,1)
     135: (3,2,2,2)         360: (3,2,2,1,1,1)
		

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 A101707.

The positive version is A101707 (A340604).

The even version is A101708 (A340930).

The not necessarily odd version is A064173 (A340788).

A001222 counts prime factors.

A027193 counts partitions of odd length (A026424).

A047993 counts balanced partitions (A106529).

A058695 counts partitions of odd numbers (A300063).

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.

A324516 counts partitions with rank equal to maximum minus minimum part (A324515).

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).

A340692 counts partitions of odd rank (A340603).

Cf. A003114, A056239, A096401, A117193, A117409, A325134, A326845, A340604, A340605, A340787, A340854/A340855.

Programs

Mathematica

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

Formula

For all terms, A061395(a(n)) - A001222(a(n)) is odd and negative.