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Search: id:A006330
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| A006330 |
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Number of corners, or planar partitions of n with only one row and one column.
(Formerly M2553)
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+0
6
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| 1, 1, 3, 6, 12, 21, 38, 63, 106, 170, 272, 422, 653, 986, 1482, 2191, 3218, 4666, 6726, 9592, 13602, 19122, 26733, 37102, 51232, 70292, 95989, 130356, 176246, 237120, 317724, 423840, 563266, 745562, 983384, 1292333, 1692790, 2209886, 2876132
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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The first four terms a(0), a(1), a(2), a(3) agree with sequence A000219 for unrestricted planar partitions, since the restriction does not rule anything out. For a(4) there is just one planar partition which doesn't satisfy the restriction, four 1's arranged in a square. So A000219 has fifth term 13, and here we have 12.
a(n)+A001523(n)=A000712(n). - Michael Somos, Jul 22 2003
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REFERENCES
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G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See (5.6).
F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs. Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
G. Kreweras, Sur les extensions lineaires d'une famille particuliere d'ordres partiels, Discrete Math., 27 (1979), 279-295.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see page 77.
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FORMULA
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G.f.: 1+Sum_{k>0} x^k/(Product_{i=1..k} (1-x^i))^2.
G.f.: (Sum_{k>=0} (-1)^k*x^(k(k+1)/2))/(Product_{k>0} 1-x^k)^2. - Michael Somos, Jul 28 2003
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polcoeff(sum(k=1, n, x^k/prod(i=1, k, 1-x^i, 1+x*O(x^n))^2, 1), n))
(PARI) a(n)=if(n<0, 0, polcoeff(sum(k=0, (sqrtint(1+8*n)-1)\2, (-1)^k*x^((k+k^2)/2))/eta(x+x*O(x^n))^2, n))
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CROSSREFS
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Cf. A000219.
Adjacent sequences: A006327 A006328 A006329 this_sequence A006331 A006332 A006333
Sequence in context: A054578 A115855 A128128 this_sequence A087503 A133627 A092176
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KEYWORD
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nonn
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AUTHOR
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njas
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EXTENSIONS
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Edited and extended by Moshe Newman (moshneumann(AT)hotmail.com), Jun 10 2003
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