decompwlj 3D

Sorting numbers: maximal number of comparisons for sorting n elements by binary insertion

A001855 on the OEIS · family summatory · also known as Sorting numbers (binary insertion)

Weight–level plate of Sorting numbers: maximal number of comparisons for sorting n elements by binary insertion
Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.
Open in the 3-D viewerA001855 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L32,017 · 32.02 %
Weight class, k ≤ L67,979 · 67.98 %
Ties, k = L46
On the level line L = 112,230
Forced level, l ≤ d²3
Range of a(n)0 … 1,568,929
Range of the jump d1 … 17
Largest weight k, level L1,568,657, 87,158

d = ceil(log2 n) exactly, constant on dyadic blocks. l/d^2 grows without bound so the level-forcing criterion never fires: 68 % weight. The gap sets a left wall - no weight column below k = d can exist.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.