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 k 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{5} S{4-5} S{3-5} S{2-5} S{1-5}
--------------------------------------------------------
. The Last Last Last The
. last two three four five
. shell shells shells shells shells
. 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
. 1 4+1 4+1 4+1 4+1
. 1 2+2+1 2+2+1 2+2+1 2+2+1
. 1 1+1 3+1+1 3+1+1 3+1+1
. 1 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
. ---------- ---------- ---------- ---------- ----------
. 8 14 17 19 20
.
So row 5 lists 8, 14, 17, 19, 20.
.
Triangle begins:
1;
2, 3;
3, 5, 6;
6, 9, 11, 12;
8, 14, 17, 19, 20;
15, 23, 29, 32, 34, 35;
19, 34, 42, 48, 51, 53, 54;
32, 51, 66, 74, 80, 83, 85, 86;
42, 74, 93, 108, 116, 122, 125, 127, 128;
64, 106, 138, 157, 172, 180, 186, 189, 191, 192;
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Apr 26 2012
STATUS
approved