A277045 Irregular triangle T(n,k) read by rows giving the number of partitions of length k such that all of the members of the partition are distinct and in A003586.

1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 2, 0, 3, 1, 1, 0, 2, 3, 1, 2, 3, 1, 0, 2, 4, 1, 0, 2, 3, 2, 0, 2, 4, 3, 1, 1, 4, 2, 1, 0, 2, 4, 3, 1, 2, 4, 4, 1, 0, 2, 5, 4, 1, 0, 3, 3, 5, 1, 0, 2, 6, 5, 2, 0, 2, 5, 5, 3, 0, 0, 7, 5, 3, 1, 2, 4, 7, 3, 1, 0, 2, 5, 8, 2, 1, 0, 2, 5, 6, 5, 1

Offset: 1

Author

Michael De Vlieger, Sep 27 2016

Keywords

• nonn
• tabf

Comments

If n is in A003586, then T(n,1) = 1, else T(n,1) = 0.

T(n,k) also is the number of ways of representing n involving k 1's in the base(2,3) or "dual-base number system" (i.e., base(2,3)).

The number of "canonic" representations of n in a dual-base number system as defined by the reference as having the lowest number of terms, appears in the first column of the triangle with a value greater than 0.

A237442(n) = the least k with a nonzero value.

Examples

			Triangle starts:
1
1
1,1
1,1
0,2
1,1,1
0,2,1
1,1,1
1,2,2
0,3,1,1
0,2,3
1,2,3,1
0,2,4,1
0,2,3,2
0,2,4,3
1,1,4,2,1
0,2,4,3
1,2,4,4,1
0,2,5,4,1
0,3,3,5,1
...
Row n = 10 has terms {0,3,1,1} because 10 is not in A003586 thus k = 1 has value 0. The partitions of 10 that have distinct members that are in A003586 are {{1,9},{2,8},{4,6},{1,3,6},{1,2,3,4}}, thus there are 3 partitions of length k = 2, 1 of length k = 3, and 1 with k = 4. A237442(10) = 2.
		

References

• V. Dimitrov, G. Jullien, R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 35-39.

Crossrefs

Cf. A003586, A237442, A276380.

Programs

Mathematica

nn = 6^6; t = Sort@ Select[Flatten@ KroneckerProduct[2^Range[0, Ceiling@ Log2@ nn], 3^Range[0, Ceiling@ Log[3, nn]]], # <= nn &]; Table[BinCounts[#, {1, Max@ # + 1, 1}] &@ Map[Length, #] &@ Select[Subsets@ TakeWhile[t, # <= n &], Total@ # == n &], {n, 40}]