The simplest coronoids: Hollow hexagons

For a positive integer d, a non-negative integer n and a non-negative integer $$h\le n$$ h ≤ n , we study the number $$C_{n}^{(d)}$$ C n ( d ) of principal ideals; and the number $$C_{n,h}^{(d)}$$ C n , h ( d ) of principal ideals generated by an element of rank h, in the d-tonal partition monoid on n elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions.

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“…Our main result in this section is the following statement which gives a direct connection between the present paper and [3,4].…”

Section: The Number Of T-hexagonsmentioning

confidence: 58%

“…Hexagonal envelopes of t-hexagons seem to be exactly the hollow hexagons considered in [3,4] (the latter papers do not really have any mathematically precise definition of hollow hexagons).…”

Section: • • • • • • • →mentioning

confidence: 94%

On the number of principal ideals in d-tonal partition monoids

2021

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“…There are only 16 hollow coronoids out of 70 primitive coronoids and 90 932 total coronoids with r < 15. 25 A protruding or intruding corner corresponds to a duo region. The number of protruding and intruding corners corresponds to η 2 and η 2 ′, respectively.…”

Section: Resultsmentioning

confidence: 99%

Structure and Electronic Characteristics of Coronoid Polycyclic Aromatic Hydrocarbons as Potential Models of Graphite Layers with Hole Defects

Dias

2008

J. Phys. Chem. A

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While the literature on coronoid hydrocarbons is comprehensively surveyed, new enumeration results on select coronoid classes are presented. The first examples of Kekulean polycyclic (even-carbon) alternant hydrocarbons with doubly degenerate zero eigenvalues at the HMO level are identified. The parallelism between a molecule removed and the antimolecule hole left in formation of a graphite vacancy defect is elaborated. It is argued that the formation of a single carbon atom vacancy hole defect in graphite should be facile because it would entail a minimum electronic reorganization. These results are supported by a substantial amount of eigenvalue calculations. This work provides a unique overview for the class of coronoid hydrocarbons.

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“…It is a purely combinatorial approach, where two given corona holes are placed, in all possible ways with respect to each other and at appropriate distances. Figures [14][15][16] show these constellations for the two smallest holes: two naphthalenes (Figure 14), phenalene and naphthalene (Figure 15), and two phenalenes (Figure 16). Notice that the number of hexagons \h) is not constant in each of the figures.…”

Section: Generating Basic Double Coronoidsmentioning

confidence: 97%