|
%I #6 Mar 31 2012 13:21:29
%S 1,3,3,3,6,6,1,3,18,3,9,24,15,3,42,38,3,10,60,69,21,6,72,153,45,6,9,
%T 114,220,141,15,1,3,120,399,274,60,3,18,159,558,570,162,12,3,174,834,
%U 1029,399,46,9,267,1080,1749,921,138,3
%N Triangle read by rows: T(n,k) counts plane partitions of n+1 that can be 'shrunk' in k ways to a plane partition of n by removing 1 element from it. Equivalently, it counts how many partitions of n+1 have k different partitions of n it just covers.
%C Sequence starts 1; 3; 3,3; 6,6,1; 3,18,3; 9,24,15; 3,42,38,3; Row sums are A000219= the plane partitions of n+1 apart from offset. Sum(all k, k * T(n,k) ) = A090984(n) by definition. First column is A007425. Row lengths are A120565. - _Franklin T. Adams-Watters_, Jun 14 2006
%e T(4,1)=2 because the only plane partitions of 4+1=5 that can be shrunk in only 1 way to plane partitions of 4 are {{5}} and {{1},{1},{1},{1},{1}}, producing {{4}} and {{1},{1},{1},{1}} respectively.
%e T(4,1)=3 because the only plane partitions of 4+1=5 that can be shrunk in only 1 way to plane partitions of 4 are {{5}},{{1,1,1,1,1}} and {{1},{1},{1},{1},{1}}, producing {{4}},{{1,1,1,1}} and {{1},{1},{1},{1}} respectively.
%t (* functions 'planepartitions' and 'coversplaneQ', see A096574 *) Table[Frequencies[Count[planepartitions[n], q_/; coversplaneQ[ #, q]]&/@ planepartitions[n+1]], {n, 1, 12}]
%Y Cf. A000219, A090984, A007425, A120565.
%K more,nonn,tabf
%O 0,2
%A _Wouter Meeussen_, Sep 12 2004
%E Corrected and extended by _Franklin T. Adams-Watters_, Jun 14 2006
%E More terms from _Wouter Meeussen_, May 05 2007
|
