A317826 Number of partitions of n with carry-free sum in factorial base.

1, 1, 1, 2, 2, 4, 1, 2, 2, 5, 4, 11, 2, 4, 4, 11, 9, 26, 3, 7, 7, 21, 16, 52, 1, 2, 2, 5, 4, 11, 2, 5, 5, 15, 11, 36, 4, 11, 11, 36, 26, 92, 7, 21, 21, 74, 52, 198, 2, 4, 4, 11, 9, 26, 4, 11, 11, 36, 26, 92, 9, 26, 26, 92, 66, 249, 16, 52, 52, 198, 137, 560, 3, 7, 7, 21, 16, 52, 7, 21, 21, 74, 52, 198, 16, 52, 52, 198, 137, 560, 31, 109

Offset: 0

Author

Antti Karttunen, Aug 08 2018

Keywords

• nonn

Comments

"Carry-free sum" in this context means that when the digits of summands (written in factorial base, see A007623) are lined up (right-justified), then summing up of each column will not result in carries to any columns left of that column, that is, the sum of digits of the k-th column from the right (with the rightmost as column 1) over all the summands is the same as the k-th digit of n, thus at most k. Among other things, this implies that in any solution, at most one of the summands may be odd. Moreover, such an odd summand is present if and only if n is odd.

a(n) is the number of set partitions of the multiset that contains d copies of each number k, collected over all k in which digit-positions (the rightmost being k=1) there is a nonzero digit d in true factorial base representation of n, where also digits > 9 are allowed.

Distinct terms are the distinct terms in A050322, that is, A045782. - David A. Corneth & Antti Karttunen, Aug 10 2018

Examples

			  n  in fact.base  a(n) carry-free partitions
------------------------------
  0     "0"         1   {}    (unique empty partition, thus a(0) = 1)
  1     "1"         1   {1}
  2    "10"         1   {2}
  3    "11"         2   {2, 1} and {3}, in fact.base: {"10", "1"} and {"11"}
  4    "20"         2   {2, 2} and {4}, in fact.base: {"10" "10"} and {"20"}
  5    "21"         4   {2, 2, 1}, {3, 2}, {4, 1} and {5},
    in factorial base:  {"10", "10", "1"}, {"11", "10"}, {"20", "1"} and {"21"}.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..5040
• Index entries for sequences related to factorial base representation
• Index entries for sequences related to partitions

Crossrefs

Cf. A001055, A007623, A025487, A045782 (range of this sequence), A050322, A276076, A278236.

Cf. A317827 (positions of records), A317828 (record values), A317829.

Cf. also A227154, A317836.

Programs

PARI

fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s};
A001055(n) = fcnt(n, n); \\ From A001055
A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A317826(n) = A001055(A276076(n));

PARI

\\ Slightly faster, memoized version:
memA001055 = Map();
A001055(n) = {my(v); if(mapisdefined(memA001055,n), v = mapget(memA001055,n), v = fcnt(n, n); mapput(memA001055,n,v); (v));}; \\ Cached version.
A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A317826(n) = A001055(A046523(A276076(n)));

Formula

a(n) = A001055(A276076(n)) = A001055(A278236(n)).

a(A000142(n)) = 1.

a(A001563(n)) = A000041(n).

a(A033312(n+1)) = A317829(n) for n >= 1.