Rewriting partitions using exponents

I’m looking for a way to re-express a partition given in full form, like
${{2, 2, 1, 1}}$, into the shortened form ${2^2, 1^2}$, i.e. given a partition with repeated entries, count the number of repetitions of a given entry, and convert this (without evaluating the $a^b$) to exponential form.

I’m aware that “Tally” will produce the correct count:

Tally({2,2,1,1})

correctly returns ${{2,2},{1,2}}$ but converting this to $2^2 1^2$ is the part that gives me trouble.

A final refinement would be that, when $1$ occurs as an exponent, it is NOT displayed, i.e. ${4,1,1}$ is displayed in shortened form as ${4, 1^2}$. The separation by “,” is optional but if the entries are in the double digits it makes a little more sense to have the “,”.

abstract algebra – “Taylor series” in the context of partitions of a number

Aluffi IV.4.4 (on the symmetric group) suggests the following exercise:

Make sense of the ‘Taylor series’ of the infinite product
$$
frac{1}{1 – x} cdot frac{1}{1 – x^2} cdot frac{1}{1 – x^3} cdot frac{1}{1 – x^4} cdot frac{1}{1 – x^5} cdot cdots.
$$

Prove that the coefficient of $x^n$ in this series is the number of partitions of $n$.

So let’s start with the first few examples:

  • $x$ is contributed by $frac{1}{1 – x}$ — that’s not particularly enlightening, so let’s keep going.
  • $x^2$ — by $frac{1}{1 – x}$ and $frac{1}{1 – x^2}$.
  • $x^3$ — by $frac{1}{1 – x}$, $frac{1}{1 – x^3}$ and $frac{1}{1 – x} cdot frac{1}{1 – x^2}$ (as $x cdot x^2$).
  • $x^4$ — by $frac{1}{1 – x}$, $frac{1}{1 – x^2}$, $frac{1}{1 – x^4}$, $frac{1}{1 – x} cdot frac{1}{1 – x^2}$ (as $x^2 cdot x^2$) and $frac{1}{1 – x} cdot frac{1}{1 – x^3}$ (as $x cdot x^3$).

At this point I’m staring at these expressions and trying to come up with a rule to match them with how I’d partition the corresponding naturals, but I’m not sure if I can make sense out of it. So how do I do that?

partitioning – Which partitions should I delete after remove Ubuntu?

My laptop has two disks: Disk 0 is SSD (Windows) and Disk 1 is HDD (Data).
Then I created a partition (Partition 16 in the picture) to install Ubuntu in Disk 2.

A few days ago, I wanted to remove Ubuntu from my computer so I cleaned the Partition 16. Now I need to reinstall Windows, I have clean the disk 0, but when I install, there is an error:

We couldn’t create a new partition or locate an existing one. For more
information, see the Setup log files.

I recognize that now there are 17 partitions in Disk 1. I think I could delete some of them but I’m afraid that I would lost the data in Partition 1, 2 and 3. I’m so confused. And if I delete some partitions, could I install Windows successfully?

Partitions_1

Partitions_2

Partitions_3

Partitions_4

List of volumes

List of disks

List of partitions in disk 1

Please help me. Thank you so much!

dual boot – Can’t see the windows partitions from Ubuntu

I have a Dell Latitude 3500 with a 1GB HDD and Window 10 on and a 500GB NVME drive which I have added and installed Ubuntu 18.04 on. Ubuntu did not see the Windows 10 installation but did see the efi boot partition and grub allows me to boot into Windows or Ubuntu.

john@john-Latitude-3500:~$ sudo df -h

Filesystem Size Used Avail Use% Mounted on
udev 3.9G 0 3.9G 0% /dev
tmpfs 786M 2.1M 783M 1% /run
/dev/nvme0n1p1 458G 32G 404G 8% /
tmpfs 3.9G 44M 3.8G 2% /dev/shm
tmpfs 5.0M 4.0K 5.0M 1% /run/lock
tmpfs 3.9G 0 3.9G 0% /sys/fs/cgroup
/dev/loop1 3.8M 3.8M 0 100% /snap/gnome-system-monitor/51
/dev/loop0 55M 55M 0 100% /snap/core18/1754
/dev/loop2 276M 276M 0 100% /snap/octave/29
/dev/loop3 13M 13M 0 100% /snap/gnome-characters/103
/dev/loop5 35M 35M 0 100% /snap/gtk-common-themes/319
/dev/loop4 15M 15M 0 100% /snap/gnome-logs/37
/dev/loop6 141M 141M 0 100% /snap/gnome-3-26-1604/70
/dev/loop7 2.4M 2.4M 0 100% /snap/gnome-calculator/180
/dev/loop8 87M 87M 0 100% /snap/core/4917
/dev/sda1 146M 46M 101M 32% /boot/efi
tmpfs 786M 16K 786M 1% /run/user/121
tmpfs 786M 28K 786M 1% /run/user/1000

gparted shows sda1 to sda4 with sda3 being the main windows partition. Can anyone tell me how to mount sda3 so that I can view it in Nautilus

Many thanks for any help

John

partitioning: is there any good software to recover deleted ext4 partitions?

I recently accidentally turned four ext4 partitions (and a linux-swap partition but it doesn't matter because it is swap space) in unallocated space, luckily I had a backup of most of my irreplaceable data. Theoretically, all my data should be there, but the drive doesn't know it.

Is there any good tool, preferably bootable, to recover such partitions?

computation models: how to find optimal subsets of partitions

Given a partition [0,1], there are 5000 subsets of partitions Pi = [ai1, ai2] ∈ [0,1], 0≤ai1≤ai2≤1, i = 1,2, …, 5000. I want to analyze these subsets and find 10 subsets that contain sixty or seventy percent of the subsets. These 10 subsets could be disjoint, even overlapping.

How to find an optimal subset of partitions

Given a partition [0,1], there are 5,000 subsets of partitions Pi = [ai1, ai2] ∈ [0,1], i = 1,2, …, 5000. I want to analyze these subsets and find some subsets to contain most subsets.

With Windows 10, how do I access files on the other partitions on my hard drive?

I had to reinstall Windows 10 on my machine, but somehow managed to install it on a smaller partition of my general hard drive. My disk has 256 GB of space, but I installed it on a 10 GB partition

Before trying to redo this, I would like to access and back up my files from the other partition. I installed a spam data recovery tool and it managed to identify my files on the machine.

So my question is, is there any free way to locate and backup my files, without using data recovery tools?

How to view all partitions on an SSD

I have an SSD with 2 partitions (volumes?) On it. I can R / W both from my WIN10 PC.
However, when I mount it on my Note10, I can only see one of the partitions.
Is there a way to see both partitions when I open file manager?

backup: restore disk utility to different partitions on the same drive

Restoring with Disk Utility seems to be the best way to copy a Time-Machine backup to another disk. When I do this, erase the drive first. This prevents me from adding a second TM backup to the drive in the same way. If I partition the drive, can I use the Disk Utility restore to copy both TM backups by sending them to different partitions?

Thank you