A187077 -id:A187077 - cp's OEIS frontend

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

Joerg Arndt, Apr 29 2013

nonn

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]

			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]]

Sage Development Team, Skew Partitions, Sage Reference Manual.

\\ 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

[SkewPartitions(n).cardinality() for n in range(16)]

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

Edited by Max Alekseyev, Dec 22 2015

A319321 a(n) is the number of d+/d- diagonally convex polyplets (polyominoes which need only touch at corners) with n cells.

Original entry on oeis.org

1, 4, 20, 108, 600, 3368, 18968, 106906, 602532, 3395402, 19131460, 107788900, 607274848

Offset: 1

David Bevan, Sep 27 2018

A polyplet is d+ [d-] diagonally convex if the intersection of its interior with any line of slope 1 [-1] through the centers of the cells is connected.

			The only tetraplets that are not d+ diagonally convex are two animals consisting of a horizontal domino and a vertical domino joined at a corner. So a(4) = A006770(4) - 2 = 108.

Adam Gruber, Java program to count lattice animals.

Cf. A006770 (fixed polyplets), A187077 (row convex polyplets), A187276 (d+/d- diagonally convex polyominoes).

cp's OEIS Frontend

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

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