%I #54 Nov 30 2013 21:30:33

%S 0,0,0,1,-1,2,-2,1,2,2,-5,2,3,3,-8,1,2,2,2,4,3,-14,2,3,3,3,2,4,4,-21,

%T 1,2,2,2,4,3,1,3,5,5,4,-32,2,3,3,3,2,4,4,1,4,3,5,6,5,-45,1,2,2,2,4,3,

%U 1,3,5,5,4,-2,2,4,4,5,3,6,6,5,-65

%N Rank of the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

%C Here the rank of a "region" is defined to be the largest part minus the number of parts (the same idea as the Dyson's rank of a partition).

%C Also triangle read by rows: T(j,k) = rank of the k-th region of the last section of the set of partitions of j.

%C The sum of every row is equal to zero.

%C Note that in some rows there are several negative terms. - _Omar E. Pol_, Oct 27 2012

%C For the definition of "region" see A206437. See also A225600 and A225610. - _Omar E. Pol_, Aug 12 2013

%F a(n) = A141285(n) - A194446(n). - Omar E. Pol, Dec 05 2011

%e In the triangle T(j,k) for j = 6 the number of regions in the last section of the set of partitions of 6 is equal to 4. The first region given by [2] has rank 2-1 = 1. The second region given by [4,2] has rank 4-2 = 2. The third region given by [3] has rank 3-1 = 2. The fourth region given by [6,3,2,2,1,1,1,1,1,1,1] has rank 6-11 = -5 (see below):

%e From _Omar E. Pol_, Aug 12 2013: (Start)

%e ---------------------------------------------------------

%e . Regions Illustration of ranks of the regions

%e ---------------------------------------------------------

%e . For J=6 k=1 k=2 k=3 k=4

%e . _ _ _ _ _ _ _ _ _ _ _ _

%e . |_ _ _ | _ _ _ . |

%e . |_ _ _|_ | _ _ _ _ * * .| . |

%e . |_ _ | | _ _ * * . | . |

%e . |_ _|_ _|_ | * .| .| . |

%e . | | . |

%e . | | .|

%e . | | *|

%e . | | *|

%e . | | *|

%e . | | *|

%e . |_| *|

%e .

%e So row 6 lists: 1 2 2 -5

%e (End)

%e Written as a triangle begins:

%e 0;

%e 0;

%e 0;

%e 1,-1;

%e 2,-2;

%e 1,2,2,-5;

%e 2,3,3,-8;

%e 1,2,2,2,4,3,-14;

%e 2,3,3,3,2,4,4,-21;

%e 1,2,2,2,4,3,1,3,5,5,4,-32;

%e 2,3,3,3,2,4,4,1,4,3,5,6,5,-45;

%e 1,2,2,2,4,3,1,3,5,5,4,-2,2,4,4,5,3,6,6,5,-65;

%e 2,3,3,3,2,4,4,1,4,3,5,6,5,-3,3,5,5,4,5,4,7,7,6,-88;

%Y Row j has length A187219(j). The absolute value of the last term of row j is A000094(j+1). Row sums give A000004.

%Y Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A206437.

%K sign,tabf

%O 1,6

%A _Omar E. Pol_, Dec 04 2011