OFFSET
1,1
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.
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
EXAMPLE
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)
MATHEMATICA
rk[n_]:=PrimePi[FactorInteger[n][[-1, 1]]]-PrimeOmega[n];
Select[Range[2, 100], OddQ[rk[#]]&&rk[#]<0&]
CROSSREFS
Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A101707.
A001222 counts prime factors.
A061395 selects the maximum prime index.
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 -
A257541 gives the rank of the partition with Heinz number n.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 29 2021
STATUS
approved