In mathematics and computer science, the sorting numbers are a sequence of numbers introduced in 1950 by Hugo Steinhaus for the analysis of comparison sort algorithms. These numbers give the worst-case number of comparisons used by both binary insertion sort and merge sort. However, there are other algorithms that use fewer comparisons.

The 👁 {\displaystyle n}
th sorting number is given by the formula^[1]

The sequence of numbers given by this formula (starting with 👁 {\displaystyle n=1}
) is

The same sequence of numbers can also be obtained from the recurrence relation^[2],

👁 {\displaystyle A(n)=A{\bigl (}\lfloor n/2\rfloor {\bigr )}+A{\bigl (}\lceil n/2\rceil {\bigr )}+n-1}
.

It is an example of a 2-regular sequence.^[2]

Asymptotically, the value of the 👁 {\displaystyle n}
th sorting number fluctuates between approximately 👁 {\displaystyle n\log _{2}n-n}
and 👁 {\displaystyle n\log _{2}n-0.915n,}
depending on the ratio between 👁 {\displaystyle n}
and the nearest power of two.^[1]

In 1950, Hugo Steinhaus observed that these numbers count the number of comparisons used by binary insertion sort, and conjectured (incorrectly) that they give the minimum number of comparisons needed to sort 👁 {\displaystyle n}
items using any comparison sort. The conjecture was disproved in 1959 by L. R. Ford Jr. and Selmer M. Johnson, who found a different sorting algorithm, the Ford–Johnson merge-insertion sort, using fewer comparisons.^[1]

The same sequence of sorting numbers also gives the worst-case number of comparisons used by merge sort to sort 👁 {\displaystyle n}
items.^[2]

The sorting numbers (shifted by one position) also give the sizes of the shortest possible superpatterns for the layered permutations.^[3]