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A259480 T(n,m) counts connected skew Ferrers diagrams of shape lambda/mu with lambda and mu partitions of n and m respectively (0<=m<=n). 11
0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 5, 1, 0, 0, 0, 7, 2, 0, 0, 0, 0, 11, 5, 2, 0, 0, 0, 0, 15, 8, 4, 0, 0, 0, 0, 0, 22, 14, 10, 3, 0, 0, 0, 0, 0, 30, 21, 18, 7, 1, 0, 0, 0, 0, 0, 42, 32, 32, 17, 6, 0, 0, 0, 0, 0, 0, 56, 45, 50, 31, 15, 2, 0, 0, 0, 0, 0, 0, 77, 65, 80, 58, 36, 11, 2, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

In contrast to A161492, which counts the same items by area and number of columns, this sequence appears to have no known generating function.

The diagonals T(n,n-k) count connected skew diagrams with weight k:

1; 2; 3,1; 5,2,2; 7,5,4,3,1; 11,8,10,7,6,2,2;

Their sums equal A006958.

REFERENCES

I. G. MacDonald: "Symmetric functions and Hall polynomials"; Oxford University Press, 1979. Page 4.

LINKS

Table of n, a(n) for n=0..90.

Wouter Meeussen, Table n,m, T(n,m) for n= 1..27

EXAMPLE

T(7,2) = 4, the pairs of partitions are ((4,3)/(2)), ((3,3,1)/(2), ((3,2,2)/(1,1)) and ((2,2,2,1)/(1,1));

the diagrams are:

x x 0 0 ,  x x 0 , x 0 0 , x 0

0 0 0      0 0 0   x 0     x 0

           0       0 0     0 0

                           0

triangle begins:

    k=0; 1  2  3  4  5  6  7

n=0;  0

n=1;  1  0

n=2;  2  0  0

n=3;  3  0  0  0

n=4;  5  1  0  0  0

n=5;  7  2  0  0  0  0

n=6; 11  5  2  0  0  0  0

n=7; 15  8  4  0  0  0  0  0

MATHEMATICA

(* see A259479 *) factor[\[Lambda]_, \[Mu]_]/; majorsweak[\[Lambda], \[Mu]]:=Block[{a1, a2, a3}, a1=Apply[Join, Table[{i, j}, {i, Length[\[Lambda]]}, {j, \[Lambda][[i]], \[Lambda][[Min[i+1, Length[\[Lambda]]]]], -1}]];

a2=Map[{First[#], First[#]>Length[\[Mu]]||\[Mu][[First[#]]]<#[[2]]}&, a1]; a3=Map[First, DeleteCases[SplitBy[a2, MatchQ[#, {_, False}]&], {{_, False}}], {2}];

Flatten[redu[Part[\[Lambda], #], Part[PadRight[\[Mu], Length[\[Lambda]], 0], #]/. 0->Sequence[]]&/@Map[Union, a3], 1]];

Table[Sum[Boole[majorsweak[\[Lambda], \[Mu]]&&redu[\[Lambda], \[Mu]]==factor[\[Lambda], \[Mu]]=={\[Lambda], \[Mu]}], {\[Lambda], Partitions[n]}, {\[Mu], Partitions[k]}], {n, 0, 12}, {k, 0, n}]

CROSSREFS

Cf. A259478, A259479, A259481, A161492, A227309, A006958.

Sequence in context: A212209 A259481 A132825 * A280164 A049597 A210951

Adjacent sequences:  A259477 A259478 A259479 * A259481 A259482 A259483

KEYWORD

nonn,tabl

AUTHOR

Wouter Meeussen, Jul 01 2015

STATUS

approved

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Last modified March 26 10:18 EDT 2019. Contains 321491 sequences. (Running on oeis4.)