A108571: Total number of terms

[Antti Karttunen]

Let's take (almost) a full circle, and consider those set partitions 
with distinct
block sizes, and then filter from the "factorial codes of all set 
partitions": (See http://www.research.att.com/~njas/sequences/A120696 )
(map A007623 seqA120696)
;Value 16: (0 1 11 21 111 121 211 221 321 1111 1121 1211 1221 1321 2111 
2121 2211 2221 2321 3121 3211 3221 3321 4321 11111 11121 11211 11221 
11321 12111 12121 12211 12221 12321 13121 13211 13221 13321 14321 21111 
21121 21211 21221 21321 22111 22121 22211 22221 22321 23121 23211 23221 
23321 24321 31121 31211 31221 31321)
those ones that encode set partitions with distinct block sizes:

So we have:
1,  (corresponds with 1 of A108571)
11, (corresponds with 22 of A108751)
111, (with 333 of A108751)
121, (with 212)
211, (with 221)
221, (with 122)
1111, (with 4444)
1121, (with 3133)
1211, (with 3313)
2111, (with 3331)
2221, (with 1333),
11111, (with 55555)
11121, (with 41444)
etc.
and 1111111111 with "TTTTTTTTTT" (not found in A108571)
(where the "T" is digit ten, i.e. with vector 
[10,10,10,10,10,10,10,10,10,10]
as we are not limited here by any arbitrary and artificial ceiling... .-)

So, maybe this makes a pair of sequences (both in decimal form
(interpret those numbers in factorial base: 1 -> 1, 11 -> 3, 111 -> 9, 
121 -> 11,
211 > 15, 221 -> 17, 1111 -> 33, to get the sequence 1,3,9,11,15,17,33,...
and the "for human consumption" version, where we see the above codes:
(map A007623 '(1 3 9 11 15 17 33))
--> (1 11 111 121 211 221 1111)



Antti Karttunen wrote:

> zak seidov wrote:
>
>> Dear seqfans,
>>
>> Anybody may wish to check these results:
>>
>> Mumber of n-digit terms in A108571: (n=1..45)
>>
>> 1,1,4,5,16,82,169,541,2272,17965,44407,201751,801515,4890886,52218595,165519640,835947970,4290442728,24096524166,179566203960,2739764737710,9938147178960,60997160143920,331360222255920,2154105076695000,14308355062630200,148898652724750500,3043362702904524000,12550859255187653400,85564729840752162000,446033694177751680000,3160644316242901488000,23904928042959835872000,212227787619709557696000,2872257514324824658032000,85739562818913709978272000,359325740171513750386752000,1764944667656072549494848000,10004773552120178696264400000,69366429961366572294099840000,546927620849236435395787200000,6125589353511448076432816640000,65850085550248066821652778880000,1448701882105457470076361135360000,65191584694745586153436251091200000. 
>>
>> Summing this, total number of terms in A108571 is: 
>> 66712890763701234740813164553708284.
>> Thanks, Zak
>
> This is actually nice:
>
> %I A108571
> %S A108571 
> 1,22,122,212,221,333,1333,3133,3313,3331,4444,14444,22333,23233,23323,
> %T A108571 
> 23332,32233,32323,32332,33223,33232,33322,41444,44144,44414,44441,
> %U A108571 
> 55555,122333,123233,123323,123332,132233,132323,132332,133223,133232
> %N A108571 Any digit d in the sequence says: "I am part of an integer 
> in which you'll find d digits "d".
> %C A108571 The sequence is finite. Last term: 
> 999999999888888887777777666666555554444333221.
> %Y A108571 Sequence in context: A085828 A081931 A043498 this_sequence 
> A105776 A044354 A044735
> %Y A108571 Adjacent sequences: A108568 A108569 A108570 this_sequence 
> A108572 A108573 A108574
> %K A108571 base,easy,fini,nonn
> %O A108571 1,2
> %A A108571 Eric Angelini (eric.angelini(AT)kntv.be), Jul 05 2005
>
> If one considers the same process in factorial base 
> http://www.research.att.com/~njas/sequences/A007623
> then (a similar) sequence can be made infinite. But then only few of 
> these terms
> can be used, because there is an additional condition that the nth 
> digit from
> the right can be at most n. So only 1, 221, 233321, 323321, 332321, 
> 333221, etc. remain.


> Alternatively, one could consider vectors of arbitrarily large integers,
> not decimal numbers with max. digit 9, and this "sequence of vectors" 
> would begin
> just like A108571, and the "number of n-element vectors matching the 
> criteria"
> would begin like your sequence. Actually, I think it should be this:
> http://www.research.att.com/~njas/sequences/A007837
> "Number of partitions of n-set with distinct block sizes."
> which begins as
> 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419,
> i.e. differs from your sequence at a(11)=44419, which is not a 
> coincidence. .-)
>