A392924 The number of partitions of n such that there exists a part p, that appears in exactly p*K/100 parts, where K is the total number of parts.

1, 1, 2, 3, 5, 9, 13, 19, 28, 40, 54, 74, 98, 129, 167, 215, 272, 344, 430, 535, 662, 815, 996, 1214, 1473, 1781, 2146, 2577, 3085, 3688, 4393, 5222, 6198, 7333, 8664, 10218, 12029, 14139, 16592, 19442, 22748, 26586, 31021, 36159, 42091, 48943, 56841

Offset

19,3

Comments

This sequence was inspired by the following puzzle from the book Mathematical Puzzles and Curiosities. If you choose the answer to this question at random, what is the probability you will be correct? A:25%, B:50%, C:60%, D:25%.

If we look at parts as percentages, the sequence counts partition such that there exists part p, such that if you choose a part uniformly at random, the part p is chosen with probability p%.

References

I. David, T. Khovanova, and Y. Shpilman, Mathematical Puzzles and Curiosities, World Scientific, 2026.

Links

Sean A. Irvine, Table of n, a(n) for n = 19..100

Example

For n = 19, the partition (10,1,1,1,1,1,1,1,1,1) has 10 parts. The part equaling 10 appears 10% of the time. Thus, this partition is counted by the sequence, implying that a(19) >= 1. For n < 19, no such partition exists. For n = 19. This is the only such partition.
For n = 21, there are two partitions with the given property: (10,2,2,1,1,1,1,1,1,1,1,1) and (10,3,1,1,1,1,1,1,1,1). Hence, a(21) = 2.

Crossrefs

Cf. A000041, A276429, A392745, A394247.

Sequence in context: A049715 A076095 A085913 * A060482 A018138 A336631

Adjacent sequences: A392921 A392922 A392923 * A392925 A392926 A392927

Keyword

nonn,changed

Author

Tanya Khovanova and PRIMES STEP senior group, May 15 2026

Status

approved