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).

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

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

Table of n, a(n) for n=1..96.

Findstat, St000183: The side length of the Durfee square of an integer partition

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