login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 9 01:30 EDT 2020. Contains 335537 sequences. (Running on oeis4.)