A307583 Position where the last of all n! permutations of { 0 .. n-1 } occurs in the digits of Pi written in base n.

2, 82, 961, 15136

Offset: 2

Author

M. F. Hasler, Apr 15 2019

Keywords

• nonn
• base
• more

Comments

By "permutation of { 0 .. n-1 }" we mean a string of n distinct digits. "The last" means the permutation which occurs for the first time later than all other permutations.

Position = k means that the string starts with the digit corresponding to the weight n^-k; e.g., the first digit after the decimal point has position 1.

Examples

			Pi written in base 2 is 11.001...[2], so the first "10" occurs at position 0 (starting with the digit of units) and "01" occurs later at position a(2) = 2.
Pi written in base 3 is 10.010211012...[3], we see that the first permutation of 0..2 to appear is "102", at position 2; then "021" at position 3, then "012" at position 7, then "201" at position 12, then "120" at position 39, and finally "210", the last partition not occurring earlier, at position 82 = a(3).
Pi written in base 4 is 3.02100333...[4]; the first permutation of 0..3 is "3012" at position 0 (starting at units digit '3'), the next distinct permutation to occur is "2031" at position 27 etc.; the last permutation not to occur earlier is "2310" at position 961 = a(4).
		

Crossrefs

Cf. A307581 (first start of any permutation of 0 .. n-1 in base-n digits of Pi).

Cf. A307582 (first occurrence of "01...(n-1)" in digits of Pi written in base n).

Cf. A068987 (occurrence of 123...n in decimal digits of Pi), A121280.

Programs

PARI

A307583(n,x=Pi,m=n^n,S=[])={for(k=n-2,oo, #Set(d=digits(x\n^-k%m,n)) < n-1 && next; #Set(d)==n || vecsort(d)==[1..n-1] || next; setsearch(S,d) && next; printf("%d: %d, ",k-n+1,Vec(d,-n));S=setunion(S,[d]);#S==n!&&return(k-n+1))}