Related to Rogers-Ramanujan Identities

[Sean A. Irvine]

Hi,

With my Managing Editor hat on, I'm starting a new monthly series of questions where I'm going to ask for enhancements to various existing sequences or groups of sequences.  These will tend to be situations where I hope that with collective effort we can improve certain unsatisfactory content.

In honor of the maintainer of the former seqfan list, my inaugural request for Sept 2025 is:

Provide an explanation for the sequences from A035401 to A035428. All 28 of these sequences have the name "Related to Rogers-Ramanujan Identities", but not much else to go on. Their author Olivier Gérard died in Sept 2024, so we are never going to get more information from the source.

Many of our long term users will already be familiar with this block of sequences, but perhaps some new eyes can discover what they actually are.

I will track these questions here:

https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements

Sean.

[Gareth McCaughan]

On 01/09/2025 22:04, Sean A. Irvine wrote:

In honor of the maintainer of the former seqfan list, my inaugural request for Sept 2025 is:

Provide an explanation for the sequences from A035401 to A035428. All 28 of these sequences have the name "Related to Rogers-Ramanujan Identities", but not much else to go on. Their author Olivier Gérard died in Sept 2024, so we are never going to get more information from the source.

In case it's useful to anyone, here are how the actual sequences begin.

(Attention conservation notice: There is nothing below that contains useful information not already immediately apparent in the sequences themselves, aside from the trivial observation that some appear to be first differences of others. But anyone wanting to look at this may find a single email easier to look at than 28 separate web pages.)

They all cite the following reference: "G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109." I have this book, and indeed page 109 of it is about the Rogers-Ramanujan identities, but it doesn't shed any obvious light on these particular sequences.

A035401: 1, 4, 14, 19, 32, 37, 45, 99, 105, 109, 118, 125, 139, 197, 202, 215, 218, 225, 239, 243, 253, 272, 510, 516, 520, 528, 531, 538, 577, 583, 588, 600, 633, 641, 657

A035402: 2, 7, 20, 23, 30, 49, 53, 57, 67, 140, 143, 153, 158, 171, 179, 195, 276, 280, 284, 303, 308, 316, 335, 340, 352, 373, 685, 688, 698, 703, 716, 721, 729, 780, 784, 793, 800, 814, 852, 856, 866

A035403: 4, 8, 12, 32, 37, 45, 68, 74, 80, 85, 97, 197, 202, 215, 218, 225, 239, 243, 253, 272, 374, 380, 387, 392, 420, 424, 428, 438, 463, 470, 484

A035404: 1, 7, 14, 19, 49, 53, 57, 67, 99, 105, 109, 118, 125, 139, 276, 280, 284, 303, 308, 316, 335, 340, 352, 373, 510, 516, 520, 528, 531, 538, 577, 583, 588, 600, 633, 641

A035405: 2, 8, 12, 20, 23, 30, 68, 74, 80, 85, 97, 140, 143, 153, 158, 171, 179, 195, 374, 380, 387, 392, 420, 424, 428, 438, 463, 470, 484, 508, 685, 688, 698, 703, 716, 721, 729, 780, 784, 793, 800, 814, 852, 856, 866, 885

A035406: 5, 10, 46, 58, 63, 70, 86, 89, 273, 287, 296, 299, 354, 361, 376, 397, 400, 404, 412

A035407: 1, 10, 15, 70, 86, 89, 100, 102, 127, 134, 376, 397, 400, 404, 412, 485, 490, 501, 511, 513, 539, 547, 565

A035408: 1, 15, 24, 27, 100, 102, 127, 134, 144, 149, 180, 185, 511, 513, 539, 547, 565, 570, 658, 661, 670, 689, 694, 730, 742, 747, 752, 768

A035409: 3, 24, 27, 31, 39, 144, 149, 180, 185, 196, 205, 210, 254, 257, 266, 689, 694, 730, 742, 747, 752, 768, 771, 887, 894, 907, 916, 925, 930, 984, 1000, 1003, 1012, 1037

A035410: 3, 5, 31, 39, 46, 58, 63, 196, 205, 210, 254, 257, 266, 273, 287, 296, 299, 354, 361, 916, 925, 930, 984, 1000, 1003, 1012, 1037, 1044

A035411: 3, 10, 5, 13, 5, 8, 54, 6, 4, 9, 7, 14, 58, 5, 13, 3, 7, 14, 4, 10, 19, 238, 6, 4, 8, 3, 7, 39, 6, 5, 12, 33, 8, 16

(comment: "Apparently the first differences of A035401.")

A035412 through A035420 appear to be likewise the first diffs of A035402 through A035410. (In each case I've only checked the first few values.)

A035421: 1, 3, 5, 8, 13, 20, 22, 31, 33, 46, 48, 50, 68, 70, 72, 98, 100, 102, 105, 140, 142, 144, 147, 196, 198, 200, 203, 208, 217, 273, 275, 277, 280, 285, 294, 374, 376, 378, 381, 386, 393, 395, 406, 509, 511, 513, 516, 521, 528, 530, 541, 557, 685, 687, 689, 692

A035422: 1, 2, 3, 5, 8, 9, 13, 14, 20, 21, 22, 31, 32, 33, 46, 47, 48, 50, 68, 69, 70, 72, 76, 98, 99, 100, 102, 105, 106, 140, 141, 142, 144, 147, 148, 153, 196, 197, 198, 200, 203, 204, 208, 209, 217, 273, 274, 275, 277, 280, 281, 285, 286, 293, 294, 374, 375, 376, 378

A035423: 1, 5, 8, 13, 20, 31, 46, 53, 68, 75, 98, 105, 110, 140, 147, 152, 196, 203, 208, 215, 273, 280, 285, 292, 374, 381, 386, 393, 404, 509, 516, 521, 528, 539, 601, 685, 692, 697, 704, 715, 730, 785, 916, 923, 928, 935, 946, 961, 1020, 1050, 1213, 1220, 1225

A035424: 1, 3, 5, 8, 13, 20, 24, 31, 35, 46, 50, 53, 68, 72, 75, 98, 102, 105, 110, 140, 144, 147, 152, 196, 200, 203, 208, 215, 239, 273, 277, 280, 285, 292, 320, 374, 378, 381, 386, 393, 404, 424, 439, 509, 513, 516, 521, 528, 539, 560, 578, 601, 685, 689, 692, 697

A035425: 2, 2, 3, 5, 7, 2, 9, 2, 13, 2, 2, 18, 2, 2, 26, 2, 2, 3, 35, 2, 2, 3, 49, 2, 2, 3, 5, 9, 56, 2, 2, 3, 5, 9, 80, 2, 2, 3, 5, 7, 2, 11, 103, 2, 2, 3, 5, 7, 2, 11, 16, 128, 2, 2, 3, 5, 7, 2, 9, 2, 15, 2, 182, 2, 2, 3, 5, 7, 2, 9, 2, 15, 2, 21, 229, 2, 3, 5, 7, 2, 9, 2, 13, 2, 2, 20, 2, 31

(these seem to be the first differences of A035421)

A035426: 1, 1, 2, 3, 1, 4, 1, 6, 1, 1, 9, 1, 1, 13, 1, 1, 2, 18, 1, 1, 2, 4, 22, 1, 1, 2, 3, 1, 34, 1, 1, 2, 3, 1, 5, 43, 1, 1, 2, 3, 1, 4, 1, 8, 56, 1, 1, 2, 3, 1, 4, 1, 7, 1, 80, 1, 1, 2, 3, 1, 4, 1, 6, 1, 1, 11, 103, 1, 1, 2, 3, 1, 4, 1, 6, 1, 1, 10, 1, 16, 128, 1, 1, 2, 3, 1, 4, 1, 6, 1, 1, 9, 1, 1, 15

(these seem to be the first differences of A035422)

A035427: 4, 3, 5, 7, 11, 15, 7, 15, 7, 23, 7, 5, 30, 7, 5, 44, 7, 5, 7, 58, 7, 5, 7, 82, 7, 5, 7, 11, 105, 7, 5, 7, 11, 62, 84, 7, 5, 7, 11, 15, 55, 131, 7, 5, 7, 11, 15, 59, 30, 163, 7, 5, 7, 11, 15, 22, 40, 35, 44

(these seem to be the first differences of A035423)

A035428: 2, 2, 3, 5, 7, 4, 7, 4, 11, 4, 3, 15, 4, 3, 23, 4, 3, 5, 30, 4, 3, 5, 44, 4, 3, 5, 7, 24, 34, 4, 3, 5, 7, 28, 54, 4, 3, 5, 7, 11, 20, 15, 70, 4, 3, 5, 7, 11, 21, 18, 23, 84, 4, 3, 5, 7, 11, 15, 7, 20, 23, 5, 131, 4, 3, 5, 7, 11, 15, 7, 21, 26, 5, 30, 163, 4, 3, 5, 7, 11, 15, 7, 15, 7, 28, 5

(these seem to be the first differences of A035424)

Page 109 of Andrews _does_ fairly clearly describe two specific sequences of numbers: (1) the number of partitions of n into parts differing by at least 2, which equals the number of partitions into parts that are 1 or 4 mod 5; (2) the number of partitions of n into parts >1 differing by at least 2, which equals the number of partitions into parts that are 2 or 3 mod 5.

The first of these begins (starting at n=0) 1,1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,17,19 and is A003114 in OEIS. The second begins (starting at n=0) 1,0,1,1,1,1,2,2,3,3,4,4,6,8,9,11,12,15,16 and is A003106 in OEIS.

--
g

[Geoffrey Caveney]

Thank you, Gareth.

I have an additional observation, which is that in sequences A035401-A035405, the sequence A003114 is cited as a Crossref (the number of partitions of n into parts 5k+1 or 5k+4), while in sequences A035406-A035410, the sequence A003106 is cited as a Crossref (the number of partitions of n into parts 5k+2 or 5k+3).

Also, I observe that in sequences A035401-A035405, certain strings of terms are repeated in different places among the various sequences. For example:

terms a(3) and a(4) of A035401 are identical to terms a(3) and a(4) of A035404; 

terms a(5)-a(7) of A035401 are identical to terms a(4)-a(6) of A035403; 

terms a(8)-a(13) of A035401 are identical to terms a(9)-a(14) of A035404; 

terms a(3)-a(5) of A035402 are identical to terms a(4)-a(6) of A035405;

terms a(6)-a(9) of A035402 are identical to terms a(5)-a(8) of A035404;

terms a(2) and a(3) of A035403 are identical to terms a(2) and a(3) of A035405;

terms a(7)-a(11) of A035403 are identical to terms a(7)-a(11) of A035405.

Geoffrey 

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[Neil Sloane]

Sean,  Those seqs (A035401 etc) go back to Dec 11 1999, when I added them to the OEOS, 1999, and in those days everything was entered by me.  I kept copious log files.  The file most relevant for A035401 seems to be HIS2.55,

which lives in my directory  /Users/njasloane/FRY/GAUSS/hisdir2

HIS stands for H'book of Integer Sequences, by the way.

Here is an email I sent to OG on Dec 9 1998

a year earlier:

Received: (from njas@localhost)

        by fry.research.att.com (8.8.7/8.8.7) id IAA20903

        for nj...@research.att.com; Wed, 9 Dec 1998 08:26:00 -0500 (EST)

Date: Wed, 9 Dec 1998 08:26:00 -0500 (EST)

From: "N. J. A. Sloane" <nj...@research.att.com>

Message-Id: <1998120913...@fry.research.att.com>

To: nj...@research.att.com

Subject: copy

To: oge...@ext.jussieu.fr

Subject: Re:

by the way, would you please send me better definitions for

the sequences A035401 etc, so i can update these entries:

%I A035401

%S A035401 1,4,14,19,32,37,45,99,105,109,118,125,139,197,202,215,218,225,239,243

,

%T A035401 253,272,510,516,520,528,531,538,577,583,588,600,633,641,657

%N A035401 Related to Rogers-Ramanujan Identities.

%R A035401 Andr76 109.

%D A035401 Contact author at address below [Note: this form of description is no

t accepted! - NJAS]

%Y A035401 Cf. A003114,A035402,A035403,A035404,A035405.

%A A035401 Olivier Gerard (oge...@ext.jussieu.fr)

%O A035401 1,2

%K A035401 nonn,part

%P A035401 OG Ref: Limit Pgf 1 mod 5

thanks

NJAS

---------------------------------

The following are some relevant emails from - or to - him:

---------------------------------------

----------------------------------

His first reply:

From oge...@ext.jussieu.fr  Wed Dec  9 09:35:07 1998

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Date: Wed, 9 Dec 1998 15:40:41 +0200

To: Neil Sloane <nj...@research.att.com>

From: Olivier Gerard <oge...@ext.jussieu.fr> 

Subject: Updated entries for A035399 - A035428...

Status: R 

Dear neil,

the reason why I did use the line << %D contact author at address below>>

was not that I did not want to explain what the sequences were: it was

because I did not have a correct definition at the time. I had in mind that

during

the time I was working in these sequences to make an article about them, if

someone

encountered them, I should encourage him to contact me so that I could explain

them to him.

Besides, It is sometimes difficult for me

to work out a good definition in english. Moreover this is sometimes a

matter of taste

and you want to be reasonably consistent with other sequences' definitions.

Remember how you refined many definitions I gave

for my sequences over the years.

I am sorry but I think the best is to clear those offending %D lines

altogether, I have not

the time to write explicit definitions for them at this moment, although I

want to do so

as soon as possible (I mean end of this week). The definitions are quite

elementary actually

but time in that case is not a quantitative problem

but qualitative: I have a lot of student term papers to evaluate and this

drains me litterally.

Hopefully, I will come up soon with a little article for the JIS about some

of them

so that this will provide a HTML link to their definition.

I understand very well also the trouble with the %P lines (not only in

these RR sequences

but in other partition sequences I sent you recently as well). For most of them

this was a lack of preparation from my part, as I could have worked out

human-readable

%C lines (notably for the sequences derived from the Gordon/Goellnitz etc...

theorems).

I will do so quite soon and this will fix the trouble.

I pass already a lot of time checking and fine tuning the sequences I send you

and at the time it looked a reasonnable compromise. Please tell me if you have

particular directions or recommendations I may not have thought about that

I could

enforce in my code and my future batches.

The reason I add these %P lines most of the time is not the pleasure to be

cryptic and

to puzzle other users of the database. This is because I usually send you

only a small

selected subset of all the sequences I generate and I need these references

to avoid

duplicating work and find my way in my files when I encounter again these

sequences

in a lookup.

This is this selection process which explains in part the reason why I did not

succeed to send you the few thousand 2D-recurrence sequences I had prepared

and tested. Each time I was forced to stop momentarily their preparation, I had

to reenter again the atmosphere, added new things, made other experiments...

I will have time this weekend to resume this at least. Hope this is not too

long

to wait.

Regards,

Olivier

---------------------------------

Further emails from the same log file:

From oge...@ext.jussieu.fr  Sat Dec 19 13:53:08 1998

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Date: Sat, 19 Dec 1998 19:58:41 +0200

To: Neil Sloane <nj...@research.att.com>

From: Olivier Gerard <oge...@ext.jussieu.fr>

Subject: 3 A-numbers

Status: R

Dear Neil,

I don't forget you.

I have almost finished preparing new defs for part sequences, etc...

I would need quickly 3 A-numbers for a message to seqfan, please.

regards,

Olivier

--------------------------------------

The next email in that file has nothing to do with Olivier, but I can't resist including it for your amusement:

From Toxi...@netscape.net  Mon Dec 21 20:45:08 1998

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Date: 21 Dec 98 17:44:11 PST

From: Toxic Biohazard <Toxi...@netscape.net>

To: nj...@research.att.com

Subject: please help me!

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Status: R

i was wondering if you have a program that will take a few numbers and figure

out the formula to generate them. i.e. 5436, 6781, and 2771 (these are just

examples) i tried to write a program to do it for me, but was never really

sure where to begin in the analysis of the numbers. any help would be greatly

appreciated. thanks for the time!

-Toxic Biohazard

--------------------------------------

The next one is getting warm:

From oge...@ext.jussieu.fr  Thu Jan  7 16:24:11 1999

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Date: Thu, 7 Jan 1999 22:30:21 +0200

To: Neil Sloane <nj...@research.att.com>

From: Olivier Gerard <oge...@ext.jussieu.fr>

Subject: Comments instead of Personal

Status: RO

Neil,

remember those sequences where I added %P lines ?

Here are corresponding %C lines for them (I recommend deleting the %P lines)

regards,

%C A035937 Case k=3,i=1 of Gordon Theorem.

%C A035938 Case k=3,i=2 of Gordon Theorem.

%C A035939 Case k=3,i=3 of Gordon Theorem.

%C A035940 Case k=4,i=1 of Gordon Theorem.

%C A035941 Case k=4,i=2 of Gordon Theorem.

%C A035942 Case k=4,i=3 of Gordon Theorem.

%C A035943 Case k=4,i=4 of Gordon Theorem.

%C A035944 Case k=5,i=1 of Gordon Theorem.

%C A035945 Case k=5,i=2 of Gordon Theorem.

%C A035946 Case k=5,i=3 of Gordon Theorem.

%C A035947 Case k=5,i=4 of Gordon Theorem.

%C A035948 Case k=5,i=5 of Gordon Theorem.

%C A035949 Case k=6,i=1 of Gordon Theorem.

%C A035950 Case k=6,i=2 of Gordon Theorem.

%C A035951 Case k=6,i=3 of Gordon Theorem.

%C A035952 Case k=6,i=4 of Gordon Theorem.

%C A035953 Case k=6,i=5 of Gordon Theorem.

%C A035954 Case k=6,i=6 of Gordon Theorem.

%C A035955 Case k=7,i=1 of Gordon Theorem.

%C A035956 Case k=7,i=2 of Gordon Theorem.

%C A035957 Case k=7,i=3 of Gordon Theorem.

%C A035958 Case k=7,i=4 of Gordon Theorem.

%C A035959 Case k=7,i=5 of Gordon Theorem.

%C A035960 Case k=7,i=6 of Gordon Theorem.

%C A035961 Case k=7,i=7 of Gordon Theorem.

%C A035962 Case k=8,i=1 of Gordon Theorem.

%C A035963 Case k=8,i=2 of Gordon Theorem.

%C A035964 Case k=8,i=3 of Gordon Theorem.

%C A035965 Case k=8,i=4 of Gordon Theorem.

%C A035966 Case k=8,i=5 of Gordon Theorem.

%C A035967 Case k=8,i=6 of Gordon Theorem.

%C A035968 Case k=8,i=7 of Gordon Theorem.

%C A035969 Case k=8,i=8 of Gordon Theorem.

%C A035970 Case k=9,i=1 of Gordon Theorem.

%C A035971 Case k=9,i=2 of Gordon Theorem.

%C A035972 Case k=9,i=3 of Gordon Theorem.

%C A035973 Case k=9,i=4 of Gordon Theorem.

%C A035974 Case k=9,i=5 of Gordon Theorem.

%C A035975 Case k=9,i=6 of Gordon Theorem.

%C A035976 Case k=9,i=7 of Gordon Theorem.

%C A035977 Case k=9,i=8 of Gordon Theorem.

%C A035978 Case k=9,i=9 of Gordon Theorem.

%C A035979 Case k=10,i=1 of Gordon Theorem.

%C A035980 Case k=10,i=2 of Gordon Theorem.

%C A035981 Case k=10,i=3 of Gordon Theorem.

%C A035982 Case k=10,i=4 of Gordon Theorem.

%C A035983 Case k=10,i=5 of Gordon Theorem.

%C A035984 Case k=10,i=6 of Gordon Theorem.

%C A035985 Case k=10,i=7 of Gordon Theorem.

%C A035986 Case k=10,i=8 of Gordon Theorem.

%C A035987 Case k=10,i=9 of Gordon Theorem.

%C A035988 Case k=10,i=10 of Gordon Theorem.

%C A035989 Case k=11,i=1 of Gordon Theorem.

%C A035990 Case k=11,i=2 of Gordon Theorem.

%C A035991 Case k=11,i=3 of Gordon Theorem.

%C A035992 Case k=11,i=4 of Gordon Theorem.

%C A035993 Case k=11,i=5 of Gordon Theorem.

%C A035994 Case k=11,i=6 of Gordon Theorem.

%C A035995 Case k=11,i=7 of Gordon Theorem.

%C A035996 Case k=11,i=8 of Gordon Theorem.

%C A035997 Case k=11,i=9 of Gordon Theorem.

%C A035998 Case k=11,i=10 of Gordon Theorem.

%C A035999 Case k=11,i=11 of Gordon Theorem.

%C A036000 Case k=12,i=1 of Gordon Theorem.

%C A036001 Case k=12,i=2 of Gordon Theorem.

%C A036002 Case k=12,i=3 of Gordon Theorem.

%C A036003 Case k=12,i=4 of Gordon Theorem.

%C A036004 Case k=12,i=5 of Gordon Theorem.

%C A036005 Case k=12,i=6 of Gordon Theorem.

%C A036006 Case k=12,i=7 of Gordon Theorem.

%C A036007 Case k=12,i=8 of Gordon Theorem.

%C A036008 Case k=12,i=9 of Gordon Theorem.

%C A036009 Case k=12,i=10 of Gordon Theorem.

%C A036010 Case k=12,i=11 of Gordon Theorem.

%C A036011 Case k=12,i=12 of Gordon Theorem.

%C A036015 Case k=2,i=1 of Gordon/Goellnitz/Andrews Theorem.

%C A036016 Case k=2,i=2 of Gordon/Goellnitz/Andrews Theorem.

%C A036017 Case k=3,i=1 of Gordon/Goellnitz/Andrews Theorem.

%C A036018 Case k=3,i=2 of Gordon/Goellnitz/Andrews Theorem.

%C A036019 Case k=3,i=3 of Gordon/Goellnitz/Andrews Theorem.

%C A036020 Case k=4,i=1 of Gordon/Goellnitz/Andrews Theorem.

%C A036021 Case k=4,i=2 of Gordon/Goellnitz/Andrews Theorem.

%C A036022 Case k=4,i=3 of Gordon/Goellnitz/Andrews Theorem.

%C A036023 Case k=4,i=4 of Gordon/Goellnitz/Andrews Theorem.

%C A036024 Case k=5,i=1 of Gordon/Goellnitz/Andrews Theorem.

%C A036025 Case k=5,i=2 of Gordon/Goellnitz/Andrews Theorem.

%C A036026 Case k=5,i=3 of Gordon/Goellnitz/Andrews Theorem.

%C A036027 Case k=5,i=4 of Gordon/Goellnitz/Andrews Theorem.

%C A036028 Case k=5,i=5 of Gordon/Goellnitz/Andrews Theorem.

%C A036029 Case k=6,i=1 of Gordon/Goellnitz/Andrews Theorem.

%C A036030 Case k=6,i=2 of Gordon/Goellnitz/Andrews Theorem.

%C A036031 Case k=6,i=3 of Gordon/Goellnitz/Andrews Theorem.

%C A036032 Case k=6,i=4 of Gordon/Goellnitz/Andrews Theorem.

%C A036033 Case k=6,i=5 of Gordon/Goellnitz/Andrews Theorem.

%C A036034 Case k=6,i=6 of Gordon/Goellnitz/Andrews Theorem.

Me  Sean, maybe those notes will explain other mysterious entries from him!?

---------

From oge...@ext.jussieu.fr  Fri Jan  8 04:54:56 1999

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Date: Fri, 8 Jan 1999 10:54:54 +0100

To: Neil Sloane <nj...@research.att.com>

From: Olivier Gerard <oge...@ext.jussieu.fr> 

Subject: A few new sequences

Status: RO

Dear Neil,

I have 80 sequences ready to be formatted.

Could I have some A-numbers please ?

Olivier

------

Content-Type: text/plain; charset="us-ascii"

Date: Fri, 8 Jan 1999 18:41:17 +0200

To: "N. J. A. Sloane" <nj...@research.att.com>

From: Olivier Gerard <oge...@ext.jussieu.fr>

Subject: Re: A few new sequences

Status: RO

At 18:56 +0200 08/01/99, N. J. A. Sloane wrote:

> plain ascii, please

>

> never use html in the table

Ok

I will compute a few more terms this evening for some

of the sequences and they will be ready saturday.

Olivier

NB:

Jason Howald is listed in the table with the following email address

JAHO...@miavx1.acs.muohio.edu

it seems to be obsolete. Have you by chance something more recent

or a classic mail address ?

NB2:

As I am quite proud of them, here is an example of what I will send you :

%I A036801

%S A036801 1,1,1,1,1,2,2,3,3,3,4,5,8,8,9,11,12,19,20,25,29,31,43,48,64,75,77,99,

%T A036801 110,150,177,183,225,249,332

%N A036801 Partitions satisfying either one of the two conditions cn(0,5) = cn(2,5) <= cn(3,5) = cn(4,5) <= cn(1,5) or cn(3,5) = cn(4,5) <= cn(0,5) = cn(2,5) <= cn(1,5).

%A A036801 Olivier Gerard (oge...@ext.jussieu.fr)

%O A036801 1,6

%K A036801 nonn,more,nice,part

%C A036801 For a given partition cn(i,n) means: the number of its parts equal to i modulo n.

I have found 47 other equivalences like this, which form, in a way, a company to

the Rogers-Ramanujan identities.

----

Olivier

---------------------------

Sean:   I am now still only in January 8 1999, so there is still almost a year's worth of emails that I have to look through to see if he answered my question.

I am 52% of the way through HIS2.55.  That is, I am at line 87889  out of 167662 lines. Unfortunately the last email in HIS2.55  is dated April 12  1999.

So I will need to do the remainder of HIS2.55 and then start on HIS2.56 which ends on Dec 31 1999.

However, there is no further mention of A035401 anywhere in the HIS2 log files.  So maybe what I have given you here is all there is!  I won't pursue it any further. 

Best regards

Neil 

Neil J. A. Sloane, Chairman, OEIS Foundation.

Also Visiting Scientist, Math. Dept., Rutgers University, 

Email: njas...@gmail.com

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAFhKorXLxgs1kwYy2ycLFkqXU-zYBTk44c8k-0qvfwDkoLFqXw%40mail.gmail.com.

[Neil Sloane]

Sean,  I added a txt file to A035401 summarizing what I found about that sequence in my log files.  Not enough, I am afraid.

Best regards

Neil 

Neil J. A. Sloane, Chairman, OEIS Foundation.

Also Visiting Scientist, Math. Dept., Rutgers University, 

Email: njas...@gmail.com

[Geoffrey Caveney]

Sean, Neil, and all,

I think I figured it out. Observe that the immediately preceding sequences in OEIS, also by Olivier Gérard, are A035399, "Limit of the position of the n-th partition without repetition in the list of all integer partitions sorted in reverse lexicographic order", and A035400, the first differences of A035399.

Also observe that the sequences are separated into sets of 5: In sequences A035401-A035405, the sequence A003114 is cited as a Crossref (the number of partitions of n into parts 5k+1 or 5k+4), while in sequences A035406-A035410, the sequence A003106 is cited as a Crossref (the number of partitions of n into parts 5k+2 or 5k+3).

I think that Gérard was providing, in A035401 for example, the "Limit of the position of the n-th partition into parts 5k+1 or 5k+4 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 1 mod 5". He then produced separate sequences for integers == 2, 3, 4, and 0 mod 5.

This is why A035401 and A035404 begin with the term 1, because for integers == 1 and 4 mod 5, the very first (reverse lexicographic) partition satisfying the first Rogers-Ramanujan identity is the integer itself. But A035402 and A035405 begin with the term 2, because for integers == 2 and 0 mod 5, it is the 2nd reverse lexicographic partition of the integer that first satisfies the identity: 7, 6+1, ...; 10, 9+1, ..., etc. And A035403 begins with the term 4, because for integers == 3 mod 5, it is the 4th reverse lexicographic partition of the integers that first satisfies the identity: for example, 8 = 6+1+1 comes after the partitions 8, 7+1, and 6+2 in the reverse lexicographic ordered list of all integer partitions of 8.

An examination of, for example, the positions of the integer partitions of 11 satisfying the first Rogers-Ramanujan identity within the reverse lexicographic ordered list of all integers partitions of 11, confirms that their positions in this list match up with the terms of A035401 :

11 (1st position in the list), 10+1, 9+2, 9+1+1 (4th position in the list), 8+3, 8+2+1, 8+1+1+1, 7+4, 7+3+1, 7+2+2, 7+2+1+1, 7+1+1+1+1, 6+5, 6+4+1 (14th position in the list!), 6+3+2, 6+3+1+1, 6+2+2+1, 6+2+1+1+1, 6+1+1+1+1+1 (19th position in the list!!), etc.

Geoffrey

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[Gareth McCaughan]

On 02/09/2025 13:54, Geoffrey Caveney wrote:
> Sean, Neil, and all,
>
> I think I figured it out.

[...]

> I think that Gérard was providing, in A035401 for example, the "Limit
> of the position of the n-th partition into parts 5k+1 or 5k+4 in the
> list of all integer partitions sorted in reverse lexicographic order,
> for integers == 1 mod 5". He then produced separate sequences for
> integers == 2, 3, 4, and 0 mod 5.

Ingenious! Curious that he thought it worth providing those but _not_
also the corresponding figures for the other things that the RR
identities say are equinumerous with them, namely the partitions whose
parts differ by at least 2. But maybe those don't have the sort of
"limiting" property the existence of these sequences depends on?

--
g

[Geoffrey Caveney]

Likewise, I suspect that sequences A035421-A035424 are "Limit of the position of the n-th partition [of a certain Rogers-Ramanujan type] in the list of all integer partitions sorted in reverse lexicographic order", where the four Rogers-Ramanujan partition types are, in some order, (1) partitions such that the adjacent parts differ by at least 2, (2) partitions such that with k parts the smallest part is at least k, (3) partitions such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2, and (4) partitions such that with k parts the smallest part is at least k+1.

A035422 is the only of the four sequences that begins 1, 2, ..., so it looks like "Limit of the position of the n-th partition such that the adjacent parts differ by at least 2 in the list of all integer partitions sorted in reverse lexicographic order".

A035423 is the only of the four sequences that begins 1, 5, ..., so it looks like "Limit of the position of the n-th partition such that with k parts the smallest part is at least k+1 in the list of all integer partitions sorted in reverse lexicographic order".

A035421 looks like "Limit of the position of the n-th partition such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 in the list of all integer partitions sorted in reverse lexicographic order", because it begins 1, 3, 5, ..., and it includes a(7) = 22, which corresponds with the partition (x-6) + 4 + 2.

A035424 looks like "Limit of the position of the n-th partition such that with k parts the smallest part is at least k in the list of all integer partitions sorted in reverse lexicographic order", because it begins 1, 3, 5, ..., and it includes a(7) = 24, which corresponds with the partition (x-6) + 3 + 3.

Geoffrey

[Gareth McCaughan]

... And of course as soon as I send that, Geoffrey points out that some
of O.G.'s other sequences _are_ in fact doing exactly that. I should
keep my mouth shut when I don't know what I'm talking about :-).

--
g

[Sean A. Irvine]

Geoffrey,

Brilliant!

I was not expecting that someone would be able to discover a solution to this in such a short time.

So far, I was able to reproduce entirely the data for A035401 based on your explanation. I simply generated partitions of 10000001 in the required order and checked the condition.

Sean.

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAFhKorVYHMQRGCFkJiM5-Tfsk6K%2BTEWm6RBhEUge1X7AvOpGKw%40mail.gmail.com.

[Ed Pegg]

I keep reading this as the Rodgers and Hammerstein sequence.  

Which is of course 16,17,16,17,17,18,16,17,16,17,17,18  



… You are 16 going on 17
Baby, it's time to think
Better beware
Be canny and careful
Baby, you're on the brink
… You are 16 going on 17
Fellows will fall in line
Eager young lads
And roués and cads
Will offer you food and wine
… Totally unprepared are you
To face a world of men
Timid and shy and scared are you
Of things beyond your ken
… You need someone older and wiser
Telling you what to do
I am 17 going on 18
I'll take care of you
… I am 16 going on 17
I know that I'm naive
Fellows I meet may tell me I'm sweet
And willingly I believe
… I am 16 going on 17
Innocent as a rose
Bachelors dandies, drinkers of brandies
What do I know of those?
… Totally unprepared am I
To face a world of men
Timid and shy and scared am I
Of things beyond my ken
… I need someone older and wiser
Telling me what to do
You are 17 going on 18
I'll depend on you

[Sean A. Irvine]

Hi,

In summary I believe Geoffrey has found the correct explanation for all these sequences.

I was able to reproduce all the current terms except for a few values toward the end of A035425 (which if it follows the pattern, should be the first differences of A035421).

I'll look to update all the corresponding OEIS entries over the next few days.

Sean.

To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAFhKorWfLydCCdLRYFm8%2BVHOhofLW1W5M_piOfLr-XBbs5z-%2Bw%40mail.gmail.com.

[Geoffrey Caveney]

Thank you, Sean.

One of the more interesting phenomena to be observed in the data in Olivier Gérard's sequences is the relationship between the growth of A035421 (limit of the position of the n-th partition such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 in the list of all integer partitions sorted in reverse lexicographic order) and the growth of A035401-405 (limit of the position of the n-th partition into parts 5k+1 or 5k+4 in the list of all integer partitions sorted in reverse lexicographic order, for integers == 1, 2, 3, 4, and 0 mod 5 respectively).

It is interesting because we know that for any given integer, the total number of the latter partition type (A003114) is greater than that of the former partition type (A003106). However, Gérard's sequences demonstrate that in reverse lexicographic order, the n-th partition of the former type (A035421) in general actually occurs before the n-th partition of the latter type (A035401-405). For example, in A035421 the 56th such partition occurs in the 692nd position, while in A035401 only 35 such partitions occur before or at the 657th position.

This phenomenon occurs because more partitions with parts differing by at least 2 tend to occur closer to the beginning of reverse lexicographic order (largest part values first), whereas more partitions with parts 5k+1 or 5k+4 tend to occur closer to the end of this order. For example, among partitions of 12 in this order, 12, 10+2, 9+3, 8+4, 7+5, and 6+4+2 all occur before 6+4+1+1, 6+1+1+1+1+1+1, 4+4+4, 4+4+1+1+1+1, 4+1+1+1+1+1+1+1+1, and 1+1+1+1+1+1+1+1+1+1+1+1. Thus, if one lists the partitions in this order, up to 6+4+2 there are more partitions of the type in A035421, even though the total number of partitions of the type in A035402 is greater.

Perhaps this phenomenon is the reason why Olivier chose to place A035421 in OEIS before A035422-424, even though this order appears not to line up with the placement of A035401-405 before A035406-410. His focus may have been on the comparison between A035421 and A035401-405. 

It is possible that the ratio of the number of terms in A035421 less than a given value, to the number of terms in each of A035401-405 less than this value, approaches phi. Compare for example the 56 terms up to 692 in A035421 with the 33 terms up to 698 in A035405. If this is true, it would form quite an interesting balance with the ratio of the total number of partitions of type A035401-405 (=A003114) to that of type A035421 (=A003106), which also appears to approach phi, but with the larger and smaller values in the ratio reversed. (See my separate post on phi and the R-R identities.)

Geoffrey

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