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Dec
12
awarded  Yearling
Dec
11
answered Chromatic Polynomial of Ladder Graph
Dec
9
comment Chromatic index of a graph with vertices of degree 3 and one of degree 2
See if this helps exwiki.org/mw/…
Dec
4
awarded  Popular Question
Nov
27
comment An inequality related to the number of binary strings with no fixed substring
@RossMillikan I've checked the conjecture for all $n$ up to $12$ and $|f| \leq n$ and it appears to be true.
Nov
27
comment An inequality related to the number of binary strings with no fixed substring
@RossMillikan Observe that $S_n(f) = S_n(\overline{f})$ where $\overline{f}$ is the string obtained by inverting the bits in $f.$ Hence you can always assume both start with $0$
Nov
27
asked An inequality related to the number of binary strings with no fixed substring
Nov
18
awarded  Popular Question
Nov
17
comment Non isomorphic graphs with equal cycle matrices
What is the cycle matrix?
Nov
16
accepted Equal parity of inversions and transpositions
Nov
16
comment Equal parity of inversions and transpositions
Yes yes, this is what I was trying to clarify. Thank you once again for your kind patience, I'll drop another 50 points in bounties to you.
Nov
16
awarded  Nice Question
Nov
16
comment Equal parity of inversions and transpositions
Hm.. Why isn't $k = 4$ since $\pi( 2) > \pi(4)$ hence $(2,4)$ is an inversion and the inversion $(2,5)$ now gets affected by the transposition $(1,4)$ even though $5$ is not between $2$ and $4$
Nov
15
comment Equal parity of inversions and transpositions
Yes that is correct. But the statement in the paragraph says to consider the values of $\rho(2),\rho(3),\rho(4)$. Shouldn't there be a $p_i,p_k$ in there? And also thanks for you quick and patient comments
Nov
15
comment Equal parity of inversions and transpositions
Hm.. Maybe I don't understand the proof then. If I apply the argument of the proof $j = 2, k = 4$ and the positions in between are $2,3,4.$ This should imply that the transposition of $(1,4)$ only possible inversions affect these positions, but affects only inversions at these positions, yet $(2,5)$ was a inversion before and after applying the transposition its not anymore
Nov
15
comment Equal parity of inversions and transpositions
Hm! Maybe my notation is confusing I meant the permutation $\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 1 & 2\end{pmatrix}$
Nov
15
comment Equal parity of inversions and transpositions
Hm.. In the first sentence of the second paragraph - given the permutation $\pi = (3 4 5 1 2)$ we have that $(2,4)$ and $(2,5)$ are its inversions. However, if we swap 2 and 4, $(2,5)$ is not an inversion anymore. Yet your argument says we need only consider inversions involving $p_j,p_k$ and an element in between. Am I missing something here?
Nov
13
accepted Minimizing a specific function over n variables
Nov
11
asked Equal parity of inversions and transpositions
Nov
11
comment If $|V(G)|=n$ and $e(G)>\frac{n}{4}\{1+\sqrt{4n-3}\}$ then $G$ contains 4-cycle
See this exwiki.org/mw/…