Timeline for Number of terms in certain polynomials over $\mathbb{F}_2$
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 22, 2014 at 1:25 | vote | accept | Richard Stanley | ||
| Jan 22, 2014 at 0:33 | answer | added | Per Alexandersson | timeline score: 1 | |
| Jan 21, 2014 at 22:30 | answer | added | Gjergji Zaimi | timeline score: 11 | |
| Jan 21, 2014 at 15:20 | answer | added | James Cranch | timeline score: 1 | |
| Jan 21, 2014 at 7:44 | comment | added | Greg Martin | Still considering the "special" partitions of $2n-1$ as given by the $p_n(s_1,\dots)$: let $N_{n,k}$ $(n\ge1,k\ge0)$ denote the number of "special" partitions of $2n-1$ into $2^{k+1}-1$ parts. Then $N_{n,0}=1$ for all $n\ge1$, and $N_{n,k}=0$ when $n<2^k$. Emperically, it seems that $N_{n,k} = \sum_{j=0}^k N_{n-2^k,j}$. If I'm not mistaken, the same recursion/initial values are satisfied by the number of ways to write $n$ as a sum of powers of $2$ where the largest element of the sum equals $2^k$. This might be a clue to the desired bijection.... | |
| Jan 21, 2014 at 7:32 | comment | added | Greg Martin | Random remark: the partition $(2n-5) + 3 + 1$ never seems to occur. | |
| Jan 21, 2014 at 7:26 | comment | added | Greg Martin | @JamesCranch: emperically, the partitions of $2n-1$ into odd parts that occur all have a number of parts which is one less than a power of $2$. For example, when $n=5$, the corresponding partitions of $9$ are $9$, $7+1+1$, $3+3+3$, and $3+1+1+1+1+1+1$. However, not all such partitions appear: $5+3+1$ is missing, and $7+3+1$ and $5+3+3$ are missing when $n=6$, while $9+3+1$ and $5+3+1+1+1+1+1$ are missing when $n=7$, for example. | |
| Jan 20, 2014 at 23:17 | comment | added | James Cranch | Interesting! If we define $|s_n| = n$, then the polynomial $p_n$ is homogeneous of degree $2n-1$. In particular, if the conjecture is true, then there is some natural subset of the partitions of $2n-1$ into odd parts which is in bijection with the partitions of $n$ into powers of $2$. | |
| Jan 20, 2014 at 21:14 | history | asked | Richard Stanley | CC BY-SA 3.0 |