[seqfan] Partition numbers and Fibonacci numbers

Peter Luschny peter.luschny at gmail.com
Sat Oct 13 23:22:32 CEST 2012


Today I was looking at this number triangle, but I could
not make sense of it.

 0:  [ 1]
 1:  [ 1,   1]
 2:  [ 2,   2,   2]
 3:  [ 3,   3,   3,   3]
 4:  [ 5,   5,   5,   5,   5]
 5:  [ 7,   8,   8,   8,   8,   8]
 6:  [11,  14,  13,  13,  13,  13,  13]
 7:  [15,  23,  22,  21,  21,  21,  21,  21]
 8:  [22,  39,  36,  35,  34,  34,  34,  34,  34]
 9:  [30,  65,  60,  57,  56,  55,  55,  55,  55,  55]
10:  [42, 109,  99,  94,  91,  90,  89,  89,  89,  89,  89]
11:  [56, 183, 164, 154, 149, 146, 145, 144, 144, 144, 144, 144]

In the first column are the partition numbers and on the
diagonal are the Fibonacci numbers. Can anybody give
or point to an interpretation of this connection?

The formal definition in `Sage´ is:

@CachedFunction
def PartToFibo(n, m):
    if n < 2: return 1
    S = 0; J = n-1; j = m
    while 0 < J:
        T = PartToFibo(J, m)
        S = S-T if (j//m)%m == 0 else S+T
        J -= j//m if j%m == 0 else j
        j += 1
    return S

for n in (0..12): [PartToFibo(n+1,m+2) for m in (0..n)]

Peter

http://oeis.org/A000041  http://oeis.org/A000045



More information about the SeqFan mailing list