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

Author

Emeric Deutsch, Feb 12 2006

Keywords

• nonn
• easy
• tabl

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)

Examples

			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;
		

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

• John Tyler Rascoe, Rows n = 1..140, flattened
• Findstat, St000183: The side length of the Durfee square of an integer partition

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.

Programs

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 *)

PARI

T_qt(max_row) = {my(N=max_row+1, q='q+O('q^N), h = sum(k=1,N, q^(k^2)/prod(j=1,k, (1-q^j)*(1-t*q^j))) ); for(i=1, N-1, print(Vecrev(polcoef(h, i))))}
T_qt(10) \\ John Tyler Rascoe, Oct 24 2024

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).