Timeline for Generating function for certain partitions (with a restriction on the Durfee square)
Current License: CC BY-SA 3.0
20 events
| when toggle format | what | by | license | comment | |
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| Apr 6, 2020 at 18:24 | vote | accept | Pablo Spiga | ||
| S Jan 10, 2016 at 15:56 | history | bounty ended | Pablo Spiga | ||
| S Jan 10, 2016 at 15:56 | history | notice removed | Pablo Spiga | ||
| Jan 10, 2016 at 14:35 | comment | added | Pietro Majer | yes, a bijection between these two classes of partitions would be a nice combinatorial proof, but I can't see it... | |
| Jan 10, 2016 at 14:03 | comment | added | Fedor Petrov | @PietroMajer you are correct! This should lead to a shorter proof than this below:) | |
| Jan 10, 2016 at 8:03 | comment | added | Pietro Majer | It seems to me that the coefficient of $x^N$ in the latter generating formula also counts the partitions of $N$ in exactly $m$ parts where the smaller lacking part is even, that is of the form $N=1+1+3+4+7\dots$ or $N=1+2+2+2+3+4+5+7\dots$. | |
| Jan 9, 2016 at 23:53 | comment | added | darij grinberg | This comment thread is a train wreck. Which formula is incorrect? | |
| Jan 9, 2016 at 18:11 | history | edited | Pablo Spiga |
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| Jan 9, 2016 at 11:52 | answer | added | Fedor Petrov | timeline score: 6 | |
| Jan 6, 2016 at 17:51 | comment | added | JMP | just spotted it's compositions! | |
| Jan 6, 2016 at 17:34 | comment | added | JMP | its $(x+\dots+x^{m-i})^i$ which is i parts of at most m-i, but i don't think i'm cross-reading it correct | |
| Jan 6, 2016 at 16:54 | comment | added | Pablo Spiga | I do not know how you get your formula: it is incorrect, check small values of $m$ and $i$. Hardy and Wright, An introduction to the Theory of Numbers, formula 19.3.2: you see that $1/(x)_m$ is the generating function for the number of partitions in at most $m$ parts (and also for the number of partitions having parts of size at most $m$). Now looking at: Andrews, The theory of Partition, Theorem 3.1, the generating function for the number of partitions in at most a parts of size at most b is $(x)_{a+b}/((x)_a(x)_b)$. From this you can deduce the one you are asking for. | |
| Jan 6, 2016 at 16:31 | comment | added | Pablo Spiga | I do not know how you get your formula, but it is incorrect: as you can check by direct inspection using small values of $m$ and $i$. I give you a reference (this will be clearer) for mine. | |
| Jan 6, 2016 at 2:04 | comment | added | JMP | how do you get number of partitions of N into exactly i parts each of size at most m−i'? i get $(x(1-x^{m-i})/(1-x))^i$. | |
| S Jan 4, 2016 at 17:59 | history | bounty started | Pablo Spiga | ||
| S Jan 4, 2016 at 17:59 | history | notice added | Pablo Spiga | Authoritative reference needed | |
| Dec 28, 2015 at 5:20 | comment | added | Pablo Spiga | yes, I know that. I am not counting all $(m-i)\times (m-i)$ partitions with given Durfee square, but only the partitions having on the right of their Durfee square at most $m-i-1$ parts. | |
| Dec 27, 2015 at 20:52 | history | edited | Pablo Spiga | CC BY-SA 3.0 |
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| Dec 27, 2015 at 20:46 | history | edited | Pablo Spiga | CC BY-SA 3.0 |
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| Dec 27, 2015 at 20:41 | history | asked | Pablo Spiga | CC BY-SA 3.0 |