# [seqfan] Re: Counting hexagonal partitions of n

Allan Wechsler acwacw at gmail.com
Fri Nov 9 01:56:23 CET 2018

```Yes, compositions is a more felicitous word, because one expects a
partition not to depend on order.

arbitrary multiplicity, had parts of multiplicity two of every size down to
some threshold, and then singleton parts of every size from that threshold
down to a second. Those are proper partitions and we probably don't have
that sequence either.

On Nov 8, 2018 7:31 PM, "Frank Adams-watters via SeqFan" <
seqfan at list.seqfan.eu> wrote:

I assume you mean "decrease by one to an ending value".

Not critical, but wouldn't "hexagonal compositions" be a better name?

-----Original Message-----
From: Allan Wechsler <acwacw at gmail.com>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Thu, Nov 8, 2018 4:17 pm
Subject: [seqfan] Re: Counting hexagonal partitions of n

I came up with a somewhat more reliable way of counting these, so I can now
present enough data to show that the proposed sequence is not in OEIS. In
parentheses, I list the actual partitions as digit strings.

a(1) = 1 (1)
a(2) = 2 (11, 2)
a(3) = 4 (111, 12, 21, 3)
a(4) = 4 (1111, 121, 22, 4)
a(5) = 6 (11111, 122, 221, 23, 32, 5)
a(6) = 7 (111111, 1221, 123, 222, 321, 33, 6)
a(7) = 7 (1111111, 1222, 2221, 232, 34, 43, 7)
a(8) = 9 (11111111, 12221, 1232, 2222, 2321, 233, 332, 44, 8)
a(9) = 12 (111111111, 12222, 12321, 1233, 22221, 234, 3321, 333, 432, 45,
54, 9)

On Thu, Nov 8, 2018 at 11:48 AM Allan Wechsler <acwacw at gmail.com> wrote:

> By "hexagonal partition" I mean an ordered partitions whose parts start at
> some value, then increase by one, up to some maximum, then stay constant
at
> that maximum, for some number of steps, then decrease to an ending value.
>
> For n=9, for example, the hexagonal partitions are
>
> 1+1+1+1+1+1+1+1+1
> 1+2+2+2+2
> 1+2+3+2+1
> 1+2+3+3
> 2+2+2+2+1
> 2+3+4
> 3+3+2+1
> 3+3+3
> 4+5
> 5+4
> 9
>
> Unless I have missed some, there are 11 possibilities, so a(9) = 11.
>
> I have collected some values in which I have fairly little confidence, but
> I think that the number of such partitions of n is not in OEIS.
>
> There is an intimate connection to A116513. Each hexagon that A116513
> counts has at least one and as many as six "readings" as a hexagonal
> partition.
>
> Can someone provide some values with more confidence than I have been able
> to do?
>

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