[seqfan] Re: Does anyone recognise these number arrays?
Brendan McKay
Brendan.McKay at anu.edu.au
Sun Apr 6 19:21:48 CEST 2014
Thanks to Ed and Edwin for their replies. Andrew and Susanne
both seem to have solved it, though it will take a while to
confirm that all my arrays arise in this manner. I’m impressed!
Cheers, Brendan.
> 10. Does anyone recognise these number arrays? (Brendan McKay)
> 11. Re: Does anyone recognise these number arrays? (W. Edwin Clark)
> 12. Re: Does anyone recognise these number arrays? (L. Edson Jeffery)
> 13. Re: Does anyone recognise these number arrays? (Andrew Weimholt)
> 14. Re: Does anyone recognise these number arrays? (Susanne Wienand)
>
>
>Message: 13
>Date: Sun, 6 Apr 2014 02:49:16 -0700
>From: Andrew Weimholt <andrew.weimholt at gmail.com>
>To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>Subject: [seqfan] Re: Does anyone recognise these number arrays?
>Message-ID:
> <CAKPToLUS+K03YKu5JnpC6s923VrMVOMUvc_A1VAp_X1R5UWDWA at mail.gmail.com>
>Content-Type: text/plain; charset=ISO-8859-1
>
>Not sure what they represent, but I figured out how to generate them...
>
>take a rectangular region of Pascal's Triangle...
>
>
>
> 1
>
>
>
> 1 1
>
>
> 1 2 1
>
>
> 1 3 3 1
>
>
> 1 4 6 4 1
>
>
> 5 10 10 5 1
>
>
> 15 20 15 6 1
>
>
> 35 35 21 7 1
>
>
> 70 56 28 8
>
>
>
> 126 84 36
>
>
> 210 120
>
>
> 330
>
>
>
>and multiply each term by the corresponding term in the
>
>flipped around version of the rectangle...
>
>
>
> 330
>
>
>
> 120 210
>
>
> 36 84 126
>
>
> 8 28 56 70
>
>
> 1 7 21 35 35
>
>
> 1 6 15 20 15
>
>
> 1 5 10 10 5
>
>
>
> 1 4 6 4 1
>
>
> 1 3 3 1
>
>
> 1 2 1
>
>
> 1 1
>
>
> 1
>
>
>
>The terms in the array behave like the terms
>
>in a "weighted" Pascal's triangle, with the weights
>
>shown in parentheses below...
>
>
> 330
>
> / \
>
> (4/11)(7/11)
>
> / \
>
> 120 210
>
> / \ / \
>
> (3/10)(7/10)(4/10)(6/10)
>
> / \ / \
>
> 36 168 126
>
> / \ / \ / \
>
> (2/9)(7/9) (3/9) (6/9) (4/9) (5/9)
>
> / \ / \ / \
>
> 8 84 168 70
>
> / \ / \ / \ / \
>
> (1/8)(7/8) (2/8)(6/8) (3/8) (5/8) (4/8)(4/8)
>
> / \ / \ / \ / \
>
> 1 28 126 140 35
>
> \ / \ / \ / \ / \
>
> (7/7)(1/7) (6/7)(2/7) (5/7) (3/7) (4/7)(5/7)(2/7)
>
> \ / \ / \ / \ / \
>
> 5 60 150 100 15
>
> \ / \ / \ / \ / \
>
> (6/6) (1/6)(5/6) (2/6) (4/6) (3/6)(3/6)(4/6)(2/6)
>
> \ / \ / \ / \ / \
>
> 15 100 150 60 5
>
> \ / \ / \ / \ / \
>
> (5/5)(1/5) (4/5) (2/5) (3/5)(3/5)(2/5)(4/5)(1/5)
>
> \ / \ / \ / \ / \
>
> 35 140 126 28 1
>
> \ / \ / \ / \ /
>
> (4/4) (1/4) (3/4) (2/4)(2/4)(3/4)(1/4)(4/4)
>
> \ / \ / \ / \ /
>
> 70 168 84 8
>
> \ / \ / \ /
>
> (3/3) (1/3) (2/3)(2/3)(1/3)(3/3)
>
> \ / \ / \ /
>
> 126 168 36
>
> \ / \ /
>
> (2/2) (1/2)(1/2)(2/2)
>
> \ / \ /
>
> 210 120
>
> \ /
>
> (1/1)(1/1)
>
> \ /
>
> 330
>
>
>
>
> 210
>
> / \
>
> (4/10)(6/10)
>
> / \
>
> 84 126
>
> / \ / \
>
> (3/9)(6/9) (4/9) (5/9)
>
> / \ / \
>
> 28 112 70
>
> / \ / \ / \
>
> (2/8)(6/8)(3/8) (5/8) (4/8)(4/8)
>
> / \ / \ / \
>
> 7 63 105 35
>
> / \ / \ / \ / \
>
> (1/7)(6/7)(2/7)(5/7) (3/7) (4/7)(5/7)(2/7)
>
> / \ / \ / \ / \
>
> 1 24 90 80 15
>
> \ / \ / \ / \ / \
>
> (6/6)(1/6)(5/6) (2/6) (4/6) (3/6)(3/6)(4/6)(2/6)
>
> \ / \ / \ / \ / \
>
> 5 50 100 50 5
>
> \ / \ / \ / \ / \
>
> (5/5)(1/5) (4/5) (2/5) (3/5)(3/5)(2/5)(4/5)(1/5)
>
> \ / \ / \ / \ / \
>
> 15 80 90 24 1
>
> \ / \ / \ / \ /
>
> (4/4)(1/4) (3/4) (2/4)(2/4)(3/4)(1/4)(4/4)
>
> \ / \ / \ / \ /
>
> 35 105 63 7
>
> \ / \ / \ /
>
> (3/3) (1/3) (2/3)(2/3)(1/3)(3/3)
>
> \ / \ / \ /
>
> 70 112 28
>
> \ / \ /
>
> (2/2) (1/2)(1/2)(2/2)
>
> \ / \ /
>
> 126 84
>
> \ /
>
> (1/1)(1/1)
>
> \ /
>
> 210
>
>
>Note that the weights beneath each term sum to 1.
>
>Andrew
>
>
>
>On Sat, Apr 5, 2014 at 4:25 PM, Brendan McKay
><Brendan.McKay at anu.edu.au>wrote:
>
>> I have some finite arrays of numbers that I am trying to identify.
>> They arise in the theory of graph reconstruction. There is
>> a whole lot of them but I can't give a general description.
>>
>> One example:
>>
>> 330
>> 120 210
>> 36 168 126
>> 8 84 168 70
>> 1 28 126 140 35
>> 5 60 150 100 15
>> 15 100 150 60 5
>> 35 140 126 28 1
>> 70 168 84 8
>> 126 168 36
>> 210 120
>> 330
>>
>> Another:
>>
>> 210
>> 84 126
>> 28 112 70
>> 7 63 105 35
>> 1 24 90 80 15
>> 5 50 100 50 5
>> 15 80 90 24 1
>> 35 105 63 7
>> 70 112 28
>> 126 84
>> 210
>>
>> Does this ring a bell for anyone? Can anyone guess what they are?
>>
>> Thanks, Brendan.
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
>------------------------------
>
>Message: 14
>Date: Sun, 6 Apr 2014 12:22:12 +0200
>From: Susanne Wienand <susanne.wienand at gmail.com>
>To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>Subject: [seqfan] Re: Does anyone recognise these number arrays?
>Message-ID:
> <CAH=KaNsSOo2WmYnX8GizCXYhxqTez77nfVMQhc_nsU9YKH+TLg at mail.gmail.com>
>Content-Type: text/plain; charset=ISO-8859-1
>
>Hello Brendan
>
>they also seem to be realated to Pascal's triangle.
>Start with row 11 column 4 of Pascal's triangle, turn it upside down and
>multiply the numbers successively by the numbers of Pascal's triangle.
>
>330 * 1
>120 * 1 210 * 1
>36 * 1 84 * 2 126 * 1
>8 * 1 28 * 3 56 * 3 70 * 1
>1 * 1 7 * 4 21 * 6 35 * 4 35 * 1
>...?
>
>I didn' test all numbers, but it seems that you get your first example.
>
>Regards
>Susanne
>
>
>
>2014-04-06 2:42 GMT+02:00 L. Edson Jeffery <lejeffery2 at gmail.com>:
>
>> Brendan,
>>
>> This may not help, but Wolfdieter Lang has submitted some triangles
>>which
>> contain some of your rows, e.g., http://oeis.org/A062145 and
>> http://oeis.org/A062196.
>>
>> Ed Jeffery
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
>------------------------------
>
>Subject: Digest Footer
>
>_______________________________________________
>SeqFan mailing list
>SeqFan at list.seqfan.eu
>http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan
>
>------------------------------
>
>End of SeqFan Digest, Vol 67, Issue 2
>*************************************
More information about the SeqFan
mailing list