OFFSET
1,2
COMMENTS
The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.
EXAMPLE
For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below we can see that the sum of all parts of the first column equals the total number of parts in each set:
--------------------------------------------------------
. S{1-5} S{2-5} S{3-5} S{4-5} S{5}
--------------------------------------------------------
. The Last Last Last The
. five four three two last
. shells shells shells shells shell
. of 5 of 5 of 5 of 5 of 5
--------------------------------------------------------
.
. 5 5 5 5 5
. 3+2 3+2 3+2 3+2 3+2
. 4+1 4+1 4+1 4+1 1
. 2+2+1 2+2+1 2+2+1 2+2+1 1
. 3+1+1 3+1+1 3+1+1 1+1 1
. 2+1+1+1 2+1+1+1 1+1+1 1+1 1
. 1+1+1+1+1 1+1+1+1 1+1+1 1+1 1
. ---------- ---------- ---------- ---------- ----------
. 20 19 17 14 8
.
So row 5 lists 20, 19, 17, 14, 8.
.
Triangle begins:
1;
3, 2;
6, 5, 3;
12, 11, 9, 6;
20, 19, 17, 14, 8;
35, 34, 32, 29, 23, 15;
54, 53, 51, 48, 42, 34, 19;
86, 85, 83, 80, 74, 66, 51, 32;
128, 127, 125, 122, 116, 108, 93, 74, 42;
192, 191, 189, 186, 180, 172, 157, 138, 106, 64;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Apr 26 2012
STATUS
approved