proposed

approved

editing

proposed

Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, ~A143773, A316413, A325134, A326845, A340604, A340605, A340828.

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

A063995/A105806 count partitions by Dyson rank.

A063995/A105806 count partitions by Dyson rank.

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

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is 0undefined.

Freeman J. Dyson, <a href="https://doi.org/10.1016/S0021-9800(69)80006-2">A new symmetry of partitions</a>, Journal of Combinatorial Theory 7.1 (1969): 56-61.

FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>

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

A001222 counts prime factors.

A061395 selects the maximum prime index.

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

Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, ~A143773, A316413, A325134, A326845, A340604, A340605, A340828.

allocated for Gus WisemanHeinz 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

1,1

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 0.

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)

Select[Range[100], PrimePi[FactorInteger[#][[-1, 1]]]<PrimeOmega[#]&]

Note: A-numbers of ranking 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).

A063995/A105806 count partitions by Dyson rank.

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).

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), with strict case A117192.

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

Cf. A003114, A006141, A039900, A096401, A117409, ~A143773, A316413, A325134, A326845, A340604, A340605, A340828.

allocated

nonn

Gus Wiseman, Jan 29 2021

approved

editing

allocated for Gus Wiseman

allocated

approved