Fantastic Patterns and Where Not to Find Them

Interesting patterns are everywhere we look, but what happens when we try to avoid patterns? A permutation is a list of numbers in a specific order. When we avoid a pattern, we try not to order those numbers in certain ways. For example, the permutation 45312 avoids the 123 pattern because no three elements in the permutation are in an increasing order. In our work, we studied the permutations that avoid two different patterns of length three. We focused on the distribution of peaks, valleys, double ascents, and double descents over these sets of permutations

Exploring a New Object: the Taumutation

We define a taumutation as an nxn grid with exactly two different points in each row and column. A well known mathematical object is the permutation, which is defined as an ordered list of the elements 1,2,3,...,n. Examples of permutations of length 4 include 1423 and 2134. By thinking of the position of an element in a permutation as an x-coordinate and setting its value to be the y-coordinate, we obtain an nxn grid with only one point in each row and column. In a way, a taumutation is two permutations plotted on the same grid. We are often interested in permutations that avoid patterns. For example, permutations that avoid the pattern 132 do not have three elements from left-to-right (not necessarily consecutive), such that the first is the smallest, the second the largest, and the third between them. The space of permutations under pattern avoiding restrictions is well-documented; however, no one has explored our new mathematical object. In our work, we find a way to count how many taumutations exist on an nxn grid when we avoid two permutations of length three within the grid

Fantastic Patterns and Where Not to Find Them

Interesting patterns are everywhere we look, but what happens when we try to avoid patterns? A permutation is a list of numbers in a specific order. When we avoid a pattern, we try not to order those numbers in certain ways. For example, the permutation 45312 avoids the 123 pattern because no three elements in the permutation are in an increasing order. In our work, we studied the permutations that avoid two different patterns of length three. We focused on the distribution of peaks, valleys, double ascents, and double descents over these sets of permutations

Exploring a New Object: the Taumutation

We define a taumutation as an nxn grid with exactly two different points in each row and column. A well known mathematical object is the permutation, which is defined as an ordered list of the elements 1,2,3,...,n. Examples of permutations of length 4 include 1423 and 2134. By thinking of the position of an element in a permutation as an x-coordinate and setting its value to be the y-coordinate, we obtain an nxn grid with only one point in each row and column. In a way, a taumutation is two permutations plotted on the same grid. We are often interested in permutations that avoid patterns. For example, permutations that avoid the pattern 132 do not have three elements from left-to-right (not necessarily consecutive), such that the first is the smallest, the second the largest, and the third between them. The space of permutations under pattern avoiding restrictions is well-documented; however, no one has explored our new mathematical object. In our work, we find a way to count how many taumutations exist on an nxn grid when we avoid two permutations of length three within the grid

Deficiency of RgpG causes major defects in cell division and biofilm formation, and deficiency of LytR-CpsAPsr family proteins leads to accumulation of cell wall antigens in culture medium by Streptococcus mutans

ABSTRACT Streptococcus mutans is known to possess rhamnose-glucose polysaccharide (RGP), a major cell wall antigen. S. mutans strains deficient in rgpG , encoding the first enzyme of the RGP biosynthesis pathway, were constructed by allelic exchange. The rgpG deficiency had no effect on growth rate but caused major defects in cell division and altered cell morphology. Unlike the coccoid wild type, the rgpG mutant existed primarily in chains of swollen, “squarish” dividing cells. Deficiency of rgpG also causes significant reduction in biofilm formation ( P &lt; 0.01). Double and triple mutants with deficiency in brpA and/or psr , genes coding for the LytR-CpsA-Psr family proteins BrpA and Psr, which were previously shown to play important roles in cell envelope biogenesis, were constructed using the rgpG mutant. There were no major differences in growth rates between the wild-type strain and the rgpG brpA and rgpG psr double mutants, but the growth rate of the rgpG brpA psr triple mutant was reduced drastically ( P &lt; 0.001). Under transmission electron microscopy, both double mutants resembled the rgpG mutant, while the triple mutant existed as giant cells with multiple asymmetric septa. When analyzed by immunoblotting, the rgpG mutant displayed major reductions in cell wall antigens compared to the wild type, while little or no signal was detected with the double and triple mutants and the brpA and psr single mutants. These results suggest that RgpG in S. mutans plays a critical role in cell division and biofilm formation and that BrpA and Psr may be responsible for attachment of cell wall antigens to the cell envelope. IMPORTANCE Streptococcus mutans , a major etiological agent of human dental caries, produces rhamnose-glucose polysaccharide (RGP) as the major cell wall antigen. This study provides direct evidence that deficiency of RgpG, the first enzyme of the RGP biosynthesis pathway, caused major defects in cell division and morphology and reduced biofilm formation by S. mutans , indicative of a significant role of RGP in cell division and biofilm formation in S. mutans . These results are novel not only in S. mutans , but also other streptococci that produce RGP. This study also shows that the LytR-CpsA-Psr family proteins BrpA and Psr in S. mutans are involved in attachment of RGP and probably other cell wall glycopolymers to the peptidoglycan. In addition, the results also suggest that BrpA and Psr may play a direct role in cell division and biofilm formation in S. mutans . This study reveals new potential targets to develop anticaries therapeutics. </jats:p

Counting Homomorphisms to K4K_4-minor-free Graphs, modulo 2

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on $4$ vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus\mathrm{P}$-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; G\"obel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most $3$, as well as a full classification for the problem of counting list homomorphisms, modulo $2$