The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A114088 Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n >= 1; 0 <= k <= n-1). 39
 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 1, 1, 4, 5, 5, 3, 2, 1, 1, 1, 5, 6, 6, 5, 3, 2, 1, 1, 1, 6, 8, 8, 7, 5, 3, 2, 1, 1, 1, 7, 10, 10, 9, 7, 5, 3, 2, 1, 1, 1, 9, 13, 13, 12, 10, 7, 5, 3, 2, 1, 1, 1, 10, 16, 17, 15, 13, 10, 7, 5, 3, 2, 1, 1, 1, 12, 20, 22, 20, 17 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS From Gus Wiseman, May 21 2022: (Start) Also the number of integer partitions of n with k parts below the diagonal. For example, the partition (3,2,2,1) has two parts (at positions 3 and 4) below the diagonal (1,2,3,4). Row n = 8 counts the following partitions: 8 71 611 5111 41111 311111 2111111 11111111 44 332 2222 22211 221111 53 422 3221 32111 62 431 3311 521 4211 Indices of parts below the diagonal are also called strong nonexcedances. (End) REFERENCES G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28). G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78). LINKS FORMULA G.f. = Sum_{k>=1} q^(k^2) / Product_{j=1..k} (1 - q^j)*(1 - t*q^j). Sum_{k=0..n-1} k*T(n,k) = A114089(n). EXAMPLE T(7,2)=3 because we have [5,1,1], [3,2,1,1] and [2,2,2,1] (the bottom tails are [1,1], [1,1] and [2,1], respectively). Triangle starts: 1; 1, 1; 1, 1, 1; 2, 1, 1, 1; 2, 2, 1, 1, 1; 3, 3, 2, 1, 1, 1; 3, 4, 3, 2, 1, 1, 1; MAPLE g:=sum(z^(k^2)/product((1-z^j)*(1-t*z^j), j=1..k), k=1..20): gserz:=simplify(series(g, z=0, 30)): for n from 1 to 14 do P[n]:=coeff(gserz, z^n) od: for n from 1 to 14 do seq(coeff(t*P[n], t^j), j=1..n) od; # yields sequence in triangular form MATHEMATICA subdiags[y_]:=Length[Select[Range[Length[y]], #>y[[#]]&]]; Table[Length[Select[IntegerPartitions[n], subdiags[#]==k&]], {n, 1, 15}, {k, 0, n-1}] (* Gus Wiseman, May 21 2022 *) CROSSREFS Cf. A114087, A114089, A115995, A116365. Row sums: A000041. Column k = 0: A003114. Weak opposite: A115994. Permutations: A173018, weak A123125. Ordered: A352521, rank stat A352514, weak A352522. Opposite ordered: A352524, first col A008930, rank stat A352516. Weak opposite ordered: A352525, first col A177510, rank stat A352517. Weak: A353315. Opposite: A353318. A000700 counts self-conjugate partitions, ranked by A088902. A115720 counts partitions by Durfee square, rank stat A257990. A352490 gives the (strong) nonexcedance set of A122111, counted by A000701. Cf. A002620, A006918, A219282, A238352, A238874, A325039, A330644, A352487. Sequence in context: A114087 A215521 A008284 * A208245 A309049 A274190 Adjacent sequences: A114085 A114086 A114087 * A114089 A114090 A114091 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Feb 12 2006 STATUS approved

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

Last modified December 5 19:57 EST 2022. Contains 358588 sequences. (Running on oeis4.)