Timeline for Inequality among domino tilings of large triomino shapes
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 29, 2021 at 15:39 | comment | added | Peter Taylor | The sequence $L_n$ for $n \ge 1$ begins [12, 128832, 1524023326720, 19791909372130292011008, 281712534342378705290655154554535936]. $I_n$ begins [13, 145601, 1765722581057, 23333526083922816720025, 336575314603876110364700686838155709]. | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Nov 4, 2015 at 3:07 | comment | added | Per Alexandersson | @DouglasZare: Yep, that's true - I went down that road, that's probably the way to reduce the problem. However, even with this reduction, it is far from straightforward. The tilings of the region near where these trapezioids glue together can't be made into an injection in a straightforward manner... | |
| Nov 4, 2015 at 2:33 | comment | added | Douglas Zare | Both regions can be expressed as a union of two congruent trapezoids with base angles of $90^\circ$ and $45^\circ$. For each subset of $n$ out of $2n$ squares on the jagged edge, there is some number of domino tilings of that trapezoid with those $n$ included. For $L_n$, you sum over the subsets of the count times the count of the complementary subset. For $I_n$, you sum over the subsets of the count times the count of the complement reversed. Perhaps it would be helpful to determine the counts for the trapezoids. | |
| Nov 4, 2015 at 1:11 | history | edited | Per Alexandersson | CC BY-SA 3.0 |
added 382 characters in body
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| Nov 3, 2015 at 23:50 | comment | added | Douglas Zare | What are the counts for the first few $n$? | |
| Nov 3, 2015 at 19:27 | comment | added | Per Alexandersson | @GerhardPaseman: Yes, one can reduce it to some sort of staircase shapes in the middle - but finding an inclusion here is tricky... | |
| Nov 3, 2015 at 18:34 | comment | added | Gerhard Paseman | It isn't clear to me, but this is how I would start. Consider the middle square and look at incomplete tilings of that square where isolated squares along two edges remain uncovered. Can you biject between the cases where the two edges are adjacent versus when they are opposite? I imagine stair steps connecting uncovered squares with their covered counterparts, and shifting the dominoes to shift the uncovered squares from one edge to another. Gerhard "You're Probably Thinking This Too" Paseman, 2015.11.03 | |
| Nov 3, 2015 at 18:15 | history | asked | Per Alexandersson | CC BY-SA 3.0 |