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 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

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Last modified April 23 20:33 EDT 2024. Contains 371916 sequences. (Running on oeis4.)