[seqfan] Request for help with mystery sequence N0320 = A213385

[Neil Sloane]

Dear SeqFans:

Request for help with mystery sequence N0320 = A213385

On June 24 1971 Richard Guy sent me (among many
other sequences) two sequences that are related to refinements of
partitions.

Take n=5 as an example. Arrange all partitions of 5 into a directed graph:
  41 -> 311
 /  \   /      \
5    \/       2111 -> 11111
 \  /    \      /
  32 -> 221

starting at n^1 and ending at 1^n

Sequence A002846 gives the number of paths fom n^1 to 1^n of length n-1.
Here we see a(5)=4.

This is described in the paper (I have a copy).
P. Erdos, R. K. Guy and J. W. Moon, On refining partitions, J. London Math.
Soc., 9 (1975), 565-570.

So far so good. But Richard also sent the sequence (N0320, was A002847, now
A213385)
1, 2, 3, 7, 15, 43, 131, 468, 1776, 7559, 34022, 166749, 853823, 4682358
for n=1..14, which he describes as "Total number of paths from
n^1 towards 1^n of all lengths 0,1,...,n-1."

I am unable to match these numbers!

Again consider n=5. There is a direct path from 5^1 to all the later nodes,
since we can refine 5^1 to (say) 2111 in one step. In fact there are direct
paths
from each node in the picture to any node to its right.
If I count all paths from 5^1 to 11111, I get 18, not 15.
For n=1,2,3,4,5, I get 1,1,2,6,18 for the total number of paths.

So there are two questions: what is the correct interpretation of Richards
sequence A213385,
and how does my interpretation continue after 1,1,2,6,18 ?

-- 
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com