

A225114


Number of skew partitions of n whose diagrams have no empty rows and columns.


1



1, 1, 3, 9, 28, 87, 272, 850, 2659, 8318, 26025, 81427, 254777, 797175, 2494307, 7804529, 24419909, 76408475, 239077739, 748060606, 2340639096, 7323726778, 22915525377, 71701378526, 224349545236, 701976998795, 2196446204672, 6872555567553, 21503836486190, 67284284442622, 210528708959146
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OFFSET

0,3


COMMENTS

A skew partition S of size n is a pair of partitions [p1,p2] where p1 is a partition of the integer n1, p2 is a partition of the integer n2, p2 is an inner partition of p1, and n=n1n2. We say that p1 and p2 are respectively the inner and outer partitions of S. A skew partition can be depicted by a diagram made of rows of cells, in the same way as a partition. Only the cells of the outer partition p1 which are not in the inner partition p2 appear in the picture. [from the Sage manual, see links]


LINKS



EXAMPLE

The a(4)=28 skew partitions of 4 are
01: [[4], []]
02: [[3, 1], []]
03: [[4, 1], [1]]
04: [[2, 2], []]
05: [[3, 2], [1]]
06: [[4, 2], [2]]
07: [[2, 1, 1], []]
08: [[3, 2, 1], [1, 1]]
09: [[3, 1, 1], [1]]
10: [[4, 2, 1], [2, 1]]
11: [[3, 3], [2]]
12: [[4, 3], [3]]
13: [[2, 2, 1], [1]]
14: [[3, 3, 1], [2, 1]]
15: [[3, 2, 1], [2]]
16: [[4, 3, 1], [3, 1]]
17: [[2, 2, 2], [1, 1]]
18: [[3, 3, 2], [2, 2]]
19: [[3, 2, 2], [2, 1]]
20: [[4, 3, 2], [3, 2]]
21: [[1, 1, 1, 1], []]
22: [[2, 2, 2, 1], [1, 1, 1]]
23: [[2, 2, 1, 1], [1, 1]]
24: [[3, 3, 2, 1], [2, 2, 1]]
25: [[2, 1, 1, 1], [1]]
26: [[3, 2, 2, 1], [2, 1, 1]]
27: [[3, 2, 1, 1], [2, 1]]
28: [[4, 3, 2, 1], [3, 2, 1]]


PROG

(Sage) [SkewPartitions(n).cardinality() for n in range(16)]
(PARI) \\ The following program is significantly faster.
{
my( C=vector(n, j, 1) );
my(m=n, z, t, ret);
while ( 1, /* for all compositions C[1..m] of n */
\\ print( vector(m, n, C[n] ) ); /* print composition */
t = prod(j=2, m, min(C[j1], C[j]) + 1 ); /* A225114 */
\\ t = prod(j=2, m, min(C[j1], C[j]) + 0 ); /* A006958 */
\\ t = prod(j=2, m, C[j1] + C[j] + 0 ); /* A059716 */
\\ t = prod(j=2, m, C[j1] + C[j] + 1 ); /* A187077 */
\\ t = sum(j=2, m, C[j1] > C[j] ); /* A045883 */
ret += t;
if ( m<=1, break() ); /* last composition? */
/* create next composition: */
C[m1] += 1;
z = C[m];
C[m] = 1;
m += z  2;
);
return(ret);
}
for (n=0, 30, print1(A225114(n), ", "));


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



