A225114 Number of skew partitions of n whose diagrams have no empty rows and columns.

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

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=n1-n2. 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

Table of n, a(n) for n=0..30.

Sage Development Team, Skew Partitions, Sage Reference Manual.

Formula

Conjectured g.f.: 1/(2 - 1/(1 - x/(1 - x/(1 - x^2/(1 - x^2/(1 - x^3/(1 - x^3/(1 - ...)))))))). - Mikhail Kurkov, Sep 03 2024

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.
A225114(n)=
{
    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[j-1], C[j]) + 1 );  /* A225114 */
\\        t = prod(j=2, m, min(C[j-1], C[j]) + 0 );  /* A006958 */
\\        t = prod(j=2, m, C[j-1] + C[j] + 0 );  /* A059716 */
\\        t = prod(j=2, m, C[j-1] + C[j] + 1 );  /* A187077 */
\\        t = sum(j=2, m, C[j-1] > C[j] );  /* A045883 */
        ret += t;
        if ( m<=1, break() ); /* last composition? */
        /* create next composition: */
        C[m-1] += 1;
        z = C[m];
        C[m] = 1;
        m += z - 2;
    );
    return(ret);
}
for (n=0, 30, print1(A225114(n), ", "));
\\ Joerg Arndt, Jul 09 2013

Crossrefs

Sequence in context: A024738 A263841 A052939 * A085839 A115239 A134915

Adjacent sequences: A225111 A225112 A225113 * A225115 A225116 A225117

Keyword

nonn

Author

Joerg Arndt, Apr 29 2013

Extensions

Edited by Max Alekseyev, Dec 22 2015

Status

approved