A350568 a(n)/n! is the average number of key comparisons required to perform an indirect sort of n records with distinct keys using a two-way merge (A. D. Woodall's mergesort).

2, 19, 130, 992, 8145, 73665, 725630, 7840280, 92297011, 1176802235, 16129154724, 236335661166, 3685509077329, 60981635041557

Offset: 2

Author

Hugo Pfoertner, Jan 09 2022

Keywords

• nonn
• more

Comments

There are six places in the Algol 60 procedure mergesort where the keys are compared. The sequence is the sum of the counts of these comparisons, taken over all n! possible orders of the records.

The following table shows the maximum and average number of key comparisons.

.

n Worst case

| A350567(n)

| | Average

| | a(n)/n!

| | | Average/

| | | (n*log(n))

2 1 1.000 0.721

3 4 3.167 0.961

4 6 5.417 0.977

5 10 8.267 1.027

6 13 11.313 1.052

7 17 14.616 1.073

8 20 17.997 1.082

9 25 21.606 1.093

10 29 25.435 1.105

11 34 29.481 1.118

12 38 33.672 1.129

13 43 37.953 1.138

14 47 42.276 1.144

15 52 46.634 1.148

References

• D. E. Knuth, The Art of Computer Programming Second Edition. Vol. 3, Sorting and Searching. Chapter 5.2.4 Sorting by Merging, Pages 164-166. Addison-Wesley, Reading, MA, 1998.

Links

• A. D. Woodall, An internal sorting procedure using a two-way merge, Algorithm 45, The Computer Journal, Volume 13, Number 1, February 1970, Algorithms Supplement, pp. 110-111.

Crossrefs

Cf. A001768, A001855, A350567, A350569.