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edited Dec 24, 2015 at 15:25

If I'm not mistaken, this simple Sage code computes $F(n)$:

def F(n): P = Posets.IntegerPartitions(n) return sum( len(P.closed_interval(p,P.top())) for p in P )

The values of $F(n)$ for $n=0,1,\dots,20$ are:

1, 1, 3, 6, 14, 26, 55, 99, 192, 340, 619, 1063, 1873, 3129, 5308, 8718, 14385, 23116, 37346, 58949, 93294

UPDATE. I've added these counts as the sequence https://oeis.org/A265947 in the OEIS.

If I'm not mistaken, this simple Sage code computes $F(n)$:

def F(n): P = Posets.IntegerPartitions(n) return sum( len(P.closed_interval(p,P.top())) for p in P )

The values of $F(n)$ for $n=0,1,\dots,20$ are:

1, 1, 3, 6, 14, 26, 55, 99, 192, 340, 619, 1063, 1873, 3129, 5308, 8718, 14385, 23116, 37346, 58949, 93294

If I'm not mistaken, this simple Sage code computes $F(n)$:

def F(n): P = Posets.IntegerPartitions(n) return sum( len(P.closed_interval(p,P.top())) for p in P )

The values of $F(n)$ for $n=0,1,\dots,20$ are:

1, 1, 3, 6, 14, 26, 55, 99, 192, 340, 619, 1063, 1873, 3129, 5308, 8718, 14385, 23116, 37346, 58949, 93294

UPDATE. I've added these counts as the sequence https://oeis.org/A265947 in the OEIS.

answered Dec 22, 2015 at 16:32

If I'm not mistaken, this simple Sage code computes $F(n)$:

def F(n): P = Posets.IntegerPartitions(n) return sum( len(P.closed_interval(p,P.top())) for p in P )

The values of $F(n)$ for $n=0,1,\dots,20$ are:

1, 1, 3, 6, 14, 26, 55, 99, 192, 340, 619, 1063, 1873, 3129, 5308, 8718, 14385, 23116, 37346, 58949, 93294