reference request – Cylindric partitions for lattice paths with a weight of binomial form

In Cylindric partitions prop.1, Gessel and Krattenthaler prove a formula for lattice paths on a cylinder

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In our particular problem, we again have paths $((P_{1},k_{1}),…,(P_{r},k_{r}))$ and the weights do satisfy $w(Se)=w(e)$ for all edges, but for the weight of the final-step we have
$$Large w_{last}(x,y,k):=left{begin{matrix}
(-1)^{y-x}binom{k}{y-x}1_{kgeq y-xgeq 0},&text{ if } kgeq 0 \ (-1)^{k}binom{x-y-1}{|k|-1}1_{xgeq y+|k|}, &text{ if } k< 0 end{matrix}right.$$

meaning that for each ith path the weight on the final edge $P_{i}(N_{i}-1)to P_{i}(N_{i}):=S^{k_{i}}v_{j}$ is given by
$$w(P_{i}(N_{i}-1), S^{k_{i}}v_{j}):=w_{last}(P_{i}(N_{i}-1), S^{k_{i}}v_{j},k_{i}).$$

I am trying to see if there is any extension of prop.1 for weights as in our particular problem. The existing proof from 1 fails at the step of $P_{1}$ and $SP_{r}$ intersecting because when we insert $k’_{r}=k_{1}-1$ and $k_{1}’=k_{r}+1$, the weights of the last step change i.e. the involution $phi$ is no longer “weight-preserving”.

I am thinking that one possible remedy is to find some prefactor like the prefactor $z^{ak_{s}^{2}/2-v_{t}k_{s}}$ in prop.1 that will turn the involution $phi$ into “weight-preserving”.

Have you seen any situations with path-weights in binomial form? How about situations where one has to find a prefactor in order to make the involution $phi$ be “weight-preserving”?

partitioning – Unable to open HDD partitions after migrating Ubuntu to SSD

I recently migrated my Ubuntu 20.04 LTS from HDD to an SSD. The previous ubuntu partition on HDD was 107 Gbytes. The SSD OS boots successfully but some how does not recognize the old HDD partitions. The HDD has partitions as follows in which sda1 being old boot and sda8 as old swap partition.

sda      8:0    0 465.8G  0 disk 
├─sda1   8:1    0 106.7G  0 part /media/user/12a213e9-14aa-4b77-bf7c-77e89f242
├─sda2   8:2    0     1K  0 part 
├─sda5   8:5    0 116.4G  0 part /media/user/7A9428429427FEEF
├─sda6   8:6    0 116.4G  0 part /media/user/98EA0A9DEA0A77B0
├─sda7   8:7    0 116.4G  0 part /media/user/26D81E7ED81E4C85
├─sda8   8:8    0   7.9G  0 part 
└─sda9   8:9    0   1.9G  0 part 

While migrating I also created swap on SSD. SSD partitions are as follows.

sdb      8:16   0 223.6G  0 disk 
├─sdb1   8:17   0 213.8G  0 part /
├─sdb2   8:18   0   9.6G  0 part [SWAP]
└─sdb3   8:19   0   200M  0 part 

While opening HDD partition, I am getting error an ‘Could not display 12a213e9-14aa-4b77-bf7c-77e89f242. The file is of unknown type’. Is this happening due to wrong mounting of HDD?

nt.number theory – Natural functions giving zero on integer partitions

I’m working with integer partitions $ eta$ of $w$, with parts of size at most $n$, so that $eta$ satisfies $sum_{i = 1}^{n} i, eta_i = w$.

I have a set of functions $c_{n,w}(eta)$ on integer partitions in computer algebra in Mathematica for a given $n$ and $w$, and I want to try to reconstruct the form of the functions for general $n$ and $w$. The functions are valued on rational numbers, ie. $c_{n,w}(eta) in mathbb{Q}$.

For a given $n$ and $w$ some of the $c_{n,w}(eta)$ are zero, and I’m trying to work out the zeros first.
For example, for some $n$ and $w$ I have a $mathbb{Z}_2$ symmetry which forces the $c_{n,w}$ to be zero, depending on whether $sum_{i = 1}^{n} eta_i$ is even or odd. So I know that in those cases, I have that

$$
c_{n,w}(eta) sim 1+(-1)^{sum_{i = 1}^{n} eta_i}.$$

or $$
c_{n,w}(eta) sim 1-(-1)^{sum_{i = 1}^{n} eta_i}.$$

I also have some other cases with $c_{n,w}(eta) = 0$, but in those cases it’s not so clear to me to see what the pattern is. I’m looking for suggestions about natural functions $c_{n,w}(eta)$ which have zeros, additional to the ones I gave above.

I’ve also thought of $$c_{n,w}(eta) = sum_{i = 1}^{n} (-1)^i eta_i,$$ and $$c_{n,w}(eta) = 1-(-1)^{f(eta)}$$ for some different functions $f$, maybe such as $f(eta) = sum_{i = 1}^{n} i^a (eta_i)^b$ for $a,b in mathbb{N}.$ Neither of those seem to work for my current problem. I also considered taking the sums up to some $m < n$ in the functional forms I gave above so that I consider only some of the parts of the partition, but that didn’t seem to help either, and I’m not really sure how natural it would be to do that either.

I’m looking for functions constructed of only addition, subtraction, multiplication and integer powers of the componenets of $eta$. (So combinatorial functions like factorial or binomial coefficients would also be fine). In the end I sum the $c_{n,w}(eta)$ over $eta$ and multiply by $prod_{i=1}^{n}a_i^{eta_i}$ to construct a polynomial with a well-defined scaling weight $w$, if that helps at all in suggesting relevant functions.

As an example, I was looking at a case with $w = 9$, $n=6$. In that case, I have all of the $c_{n,w}(eta) = 0$ when $sum_{i = 1}^{n} eta_i$ is even, so I know I need the relevant function above as a factor (I also have the $mathbb{Z}_2$ symmetry in this case). I was surprised to find that there are also two additional zeros, when $eta = (3,0,2,0,0,0)$ and when $eta = (1,1,0,0,0,1)$, so I think I need to multiply in an additional factor which is zero on these two $eta$, non-zero on $eta$ where $sum_{i = 1}^{n} eta_i$ is odd, and which I don’t have any information about when $sum_{i = 1}^{n} eta_i$ is even. Does this give enough information for someone to suggest a relevant function in this particular situation?

I appreciate that this is unlikely to fully specify the function that I’m interested in. My philosophy is to try some of the more simple possible such functions, and see if they match against the form I need in Mathematica. Hopefully this way I’ll be able to reconstruct the full functional form that I’m looking for, and this should make it easier to go back and prove that form later on. (Also I enjoy doing it this way!)

Any references which may be relevant for me to read about this would also be very welcome. I have some basic understanding of integer partitions, but I’ve not worked with them before.

Formatted micro SDHC card but partitions and files remain

I have 2 Samsung EVO micro SDHC cards that I was using in a Raspberry Pi. I’m now trying to format them on my Macbook Air running Big Sur on Apple Silicon and have tried two methods for formatting, and they ran without errors but the files and partitions remain:

1 – SD Card Formatter – tried both quick format and overwrite format

2 – sudo diskutil partitionDisk /dev/diskN 1 MBR "Free Space" "%noformat%" 100% (diskN replaced with disk3 in my case)

Does anyone have any suggestions? I’m hoping I don’t have to throw them out and order new ones.

security – Problem with a sdcard under Android 7.1.2 three partitions can not be mounted / used

Happy New Year

there is a problem with my sdcard under Android 7.1.2 (Custom Rom) (root=true, adb=possible). The vFat Partition can be used, but the rest is not mounted and can not be used.

mmcblk1p1 vfat 183 GB Status : OK

mmcblk1p2 ext4 16 MB android meta Status: not mounted

mmcblk1p3 ext4 64 GB Status: not mounted (primary) (want to use that partition as adoptable Partition for extra internal memory)

mmcblk1p4 ext4 2 GB Status: not mounted (primary) (want to use that partition as SWAP Partition for extra RAM)

Trying to mount the Partition3 (64 GB) gives the error “mount: mounting /dev/block/mmcblk1p3 on /data/sdwyt2 failed: Invalid argument”
i have the apps “Apps2SD Pro” & “Link2SD” but they can’t mount the 3 partitions too.
Does anyone know how i can mount and use the three ext4 partitions?

How to tell which partitions to backup in recovery (TWRP)?

I have TWRP as my recovery on a Galaxy phone. There are so many options to choose which partitions to backup. What do they all do? My goal is to be able to recover from the largest number of disasters. However I think restoring some partitions can lead to a boot loop, because this recently happened to me.

windows – Google Drive on multiple partitions

I like to use the official Google Drive Backup & Sync software, however I have three partitions which I can’t merge together. (1) 4TB (2) 4TB (3) 2TB. I installed Google Drive on Disk 1, but I have much more space.

I like to have some folder be stored on other disks else I will be capped at 4TB while I actually have more. Is symlink a possibility, or is there another way?

partitions – Partitioning a list to several depths

I an aware of the Partition command which partitions a list into sublists. I’m curious as to whether there is an efficient way to partition a list several times over in one step. For example, can I turn the list

{1,2,3,4,5,6,7,8}

into the list

{{{1,2},{3,4}},{{5,6},{7,8}}}

through one use of the Partition command instead of two (or indeed any command). In actual fact, I’ll be going to a list of many levels of nested lists, which is why I’m keen to know the answer. Thanks in advance for any help.

nt.number theory – Number of cyclically symmetric transpose complement plane partitions in a box against a determinant

The ubiquitous Catalan numbers can be given by $C_0:=1$ and $C_{n+1}=sum_{i+j=n}^nC_iC_j$.
In the same spirit, one may define the sequence $T_0:=1$ and $T_{n+1}=sum_{i+j+k=n}T_iT_jT_k$ listed on OEIS. Keep in mind that $C_n=frac1{n+1}binom{2n}n$ while $T_n=frac1{2n+1}binom{3n}n$. They have a shared/similar property: $C_n=frac1{2n+1}binom{2n+1}n$ and $T_n=frac1{3n+1}binom{3n+1}n$ enumerate $(n,n+1)$-core partitions and $(n,2n+1)$-core partitions here is a quick reference on the topic.

If $M_n=left(C_{i+j}right)_{i,j=0}^{n-1}$, it is known that $det(M_n)=1$. After exploring the determinant of $P_n=left(T_{i+j}right)_{i,j=0}^{n-1}$, I stumbled on the values
$$det(P_n)=sum_{k=0}^{n-1}frac{(3k+1) (6k)! (2k)!}{(4k)! (4k+1)!}.tag1$$
A quick check with OEIS reveals this sequence. Hence I get prompted to ask:

QUESTION. what is the combinatorial connection between $det(P_n)$ and the “number of cyclically symmetric transpose complement plane partitions in a $(2ntimes 2n times 2n)$-box”?

REMARK. Mills-Robbins-Rumsey use the determinant of the matrix
$$left(sum_{t=0}^{2n-2}binom{i}{t-i}binom{j}{2j-t}right)_{i,j=0}^{n-1}.$$

group theory – Isomorphisms and Partitions of Permutation

Let $X$ be a finite set, and define $G$ to be the symmetric group $S_X$.

Now, for any $A subset X$, we define $B$ as the complement of $A$ in $X$, i.e. $B = X setminus A$. Define $H = {g in G mid g(A) = A }$, i.e. the subgroup of permutations of $G$ which fix the elements in $A$.

Show that $H cong S_A times X_B$.

Intuitively, I can see this being true, in that $A, B$ partition $X$, and so for any $h in H$, it’s clear that the “part of” $h$ permuting $A$ is isomorphic to $S_A$ and the “rest of” $h$ is isomorphic to $S_B$. This is quite a hand-wavy argument, and I’m wondering if there’s a way to formalize it.