[seqfan] Re: "Types/lengths of runs" in binary numbers

[franktaw at netscape.net]

Do you mean to have types of runs include which digit is being 
repeated, or is only the run length significant?  For example, is 
11000110 the same as or different from 11100100?

If length is all that is significant, you can refer to "the multiset of 
run lengths"; this expresses your idea clearly and succinctly.  It 
might be noted that this multiset is a partition.

If the digit is significant, you can define the run type to be the pair 
of run length and digit, and then refer to the multiset of run types.

Franklin T. Adams-Watters

-----Original Message-----
From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>

I said: 'By "types" of runs, it is meant that..'
I probably should have said 'by "a combination of the lengths of runs", 
it is
meant that..', or something a little more accurate as well as clear.

Sorry, English is my second language. Babbling incoherently is my 
first....

Leroy

--- On Tue, 9/1/09, Leroy Quet <q1qq2qqq3qqqq at yahoo.com> wrote:

> I have already posted several
> sequences involving the concept of "types of runs" in a
> binary number.
>
> I just posted this today.
>
>
> %I A164953
> %S A164953 1,1,1,1,2,1,1,3,2,1,1,3,4,3,1,1,4,5,6,3,1
> %N A164953 Square array read by antidiagonals: a(m,n) = the
> number of different combinations of types of runs in the
> binary representations of positive integers that contain
> exactly m 0's and n 1's in binary. (The leftmost digit must
> be 1 in each binary number.)
> %C A164953 The top row of the array is where m=0. The
> leftmost column of the array is where n=1.
> %C A164953 Clarification regarding the definition: Each
> positive integer can be thought of as a finite binary string
> with 1 as the leftmost digit. The "runs" alternate between
> those completely of 1's and those completely of 0's. Each
> run of digit b (0 or 1) is bounded by the digit 1-b or by
> the edge of the string. By "types" of runs, it is meant that
> the lengths of the runs of digit b's (b=0 or 1) form a
> permutation of the lengths of the runs of b's in all binary
> number with the same types of runs. (See example.)
> %e A164953 Consider those binary numbers with exactly four
> 1's and two 0's. There are 10 such binary numbers that each
> have a 1 as the leftmost digit. These binary numbers,
> grouped by those numbers with the same types of runs, are:
> (111100), (111010, 101110), (111001, 100111), (110110),
> (110101, 101101, 101011), (110011). There are 6 such
> groupings, so a(2,4) = 6.
> %K A164953 base,more,nonn,tabl
> %O A164953 0,5
>
> I know that the "clarification" is somewhat confusing.
> Perhaps there is some better terminology or wording that
> someone could suggest for these types of sequences.
> (Also, I probably should have had "lengths of runs" in the
> definition instead of "types of runs", I realize now.
>
>
> Second, also interesting is the sequence of the sums of the
> antidiagonals, the number of different combinations of
> lengths of runs in the binary representations of positive
> integers that each contain exactly n digits in binary.
> The first couple terms are: 1,2,4,7,12,20. Unfortunately,
> this matches up with a number of pre-existing sequences. So
> I haven't submitted this.
>
> And as always I ask, have I made any mistakes in
> calculating the sequence so far? (I figured these terms by
> hand.)
>
>
> Thanks,
> Leroy Quet