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Jan
15
comment how to count possible planar bipartitions?
Ah. At present the question says "any bipartition at all is a valid solution", so if you really want each half of the partition to be connected you should edit the text to clarify that.
Jan
14
comment how to count possible planar bipartitions?
It seems to me that the number of ways a graph can be bipartitioned is $2^{|V|-1}$ unless you add some constraints. Have I missed a constraint?
Jan
3
comment How to denote sum over partitions?
@ruadan, it depends. If you're just referring to frequencies and total then you can use the standard notation for the frequency representation and subscript the sum with $(1^{a_1}2^{a_2}\ldots)\vdash n$. If you're referring to both parts and frequencies you need to consider whether to subscript as $\lambda = (1^{a_1}2^{a_2}\ldots)\vdash n$ or just to explain in the accompanying text how the $\lambda_i$ relate to the $a_j$.
Dec
21
comment A combinatorial game theory problem
What do you mean by "vicinal squares"? Does that cover diagonally adjacent, orthogonally adjacent, or both? And why do you call this a combinatorial game theory problem? It seems to be a single combinatorial question.
Dec
10
comment How to solve this tough recurrence relation?
Cross-post on MO
Nov
27
comment Give the generator polynomial of a binary cyclic [9, 2] code.
FWIW I checked the factors of $x^9-1$ over $GF(2)$ and they're correct.
Nov
25
comment Secret Santa Perfect Loop problem
Why is $A\rightarrow B\rightarrow C\rightarrow D\rightarrow A$ the only perfect loop? What's wrong with e.g. $A\rightarrow C\rightarrow B\rightarrow D\rightarrow A$?
Nov
15
comment Upper bound for the widest matrix with no two subsets of columns with the same vector sum
Related
Nov
15
comment Upper bound for the widest matrix with no two subsets of columns with the same vector sum
It's not actually the case that "any column cannot be the linear combination of any 2 other columns" - if it were then the upper bound would be 1. The point is that general linear combination allows multiples other than 0 or 1, whereas property X does not.
Nov
12
comment Counting all possible legal board states in Quoridor
Sketch of an approach: augment the previous approach by a set of variables which track whether the cells above and below a row of intersections are reachable from the top row and from the bottom row, and another which tracks totals. This will increase the number of nodes in the graph by a factor of on the order of tens of thousands, but the number of edges won't increase by nearly such a big factor, so it should still be a feasible calculation.
Nov
8
comment Counting all possible board positions in Quoridor
@lameK, if you're happy with the answer then there's a tick-in-a-circle button which you can use to mark it as accepted. That takes it out of the "Unanswered questions" list, and will be appreciated by people looking for genuinely unanswered questions. As for tougher Quoridor questions, in principle I'm interested but I always have a few things on the go and I can't guarantee a quick response. If you want to discuss by e-mail then I have a catch-all for anything to cheddarmonk.org.
Nov
4
comment Counting all possible board positions in Quoridor
@lameK, that's right. There are 3344 ways to order one row.
Oct
31
comment Polynomial factorisation - absolute value of coefficients
Ah, that explains it. Oops.
Oct
31
comment Polynomial factorisation - absolute value of coefficients
You could also have $x^4+x^3-x^2-1 = (x-1)(x^3+2x^2+x+1)$. If I'm understanding arxiv.org/abs/0904.3057 correctly it claims that this is the unique quartic with height 1 and an irreducible factor of height greater than 1, in which case your example is a correction.
Oct
31
comment Does $A193201$ count the partitions of $n$ of arbitrary dimension?
I'm not quite sure what I'm supposed to be looking at in the example you give of partitions of 4. For fixed layout, the preformatted text (the button that looks like {} in the editor, or the result of prepending 4 spaces to each line) is a bit easier than MathJAX.
Oct
30
comment Math puzzle: 10 digit strings generations
Independent set problem
Oct
30
comment Math puzzle: 10 digit strings generations
Not quite practical as a computational solution, but this can be reduced to the independent set problem on a graph of $10!$ nodes and $\frac{10! 7!}{2}$ edges.
Oct
15
comment Number of unlabeled simple graphs with $n$ nodes even for all $ n\ge 5$?
Ah, it doesn't have simple in the title so my search didn't find it. Thanks for adding the link.
Oct
15
comment Number of unlabeled simple graphs with $n$ nodes even for all $ n\ge 5$?
Are you referring to A005470 (in which case you seem to be missing the very important word planar from the question) or to a different sequence?
Oct
12
comment Counting all possible board positions in Quoridor
@lameK, I got 55 by writing a short computer program, but as I used an obscure language to do it I didn't see any benefit to sharing it. I'll add some examples to the last paragraph.