A116365 -id:A116365 - cp's OEIS frontend

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

Emeric Deutsch, Feb 12 2006

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)

			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;

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).

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

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.

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

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 *)

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

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).

A114089 Total number of parts in the tails below the Durfee squares of all partitions of n.

Original entry on oeis.org

0, 1, 3, 6, 11, 19, 31, 50, 76, 116, 169, 247, 349, 494, 682, 941, 1274, 1724, 2296, 3054, 4014, 5263, 6833, 8854, 11373, 14578, 18556, 23561, 29736, 37447, 46903, 58619, 72925, 90518, 111899, 138044, 169665, 208111, 254436, 310456, 377687, 458625

Offset: 1

Emeric Deutsch, Feb 12 2006

nonn

			a(4) = 6 because the bottom tails of the five partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], are { }, [1], { }, [1,1] and [1,1,1], respectively, having a total of 6 parts.

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).

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A115994, A115995, A114087, A116365, A114088.

g:=sum(z^(k^2)/product((1-z^j)*(1-t*z^j),j=1..k),k=1..10): dgdt1:=simplify(subs(t=1,diff(g,t))): dgdt1ser:=series(dgdt1,z=0,55): seq(coeff(dgdt1ser,z,n),n=1..45);
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(j*b(n-j, j), j=1..n) -add(add(b(k, d)*b(n-d^2-k, d),
                         k=0..n-d^2)*d, d=1..floor(sqrt(n))):
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 09 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Sum[j*b[n-j, j], {j, 1, n}] - Sum[Sum[b[k, d]*b[n-d^2-k, d], {k, 0, n-d^2}]*d, {d, 1, Floor[Sqrt[n]]}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

a(n) = Sum_{k=0..n-1} k*A114088(n,k).
G.f.: [(d/dt){sum(q^(k^2)/product((1-q^j)(1-tq^j), j=1..k), k=1..infinity)}]_{t=1}.
a(n) = A006128(n) - A115995(n). - Vladeta Jovovic, Feb 18 2006

A114087 Triangle read by rows: T(n,k) is the number of partitions of n whose tails below their Durfee squares have size k (n>=1; 0<=k<=n-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 2, 3, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 4, 3, 5, 3, 4, 1, 1, 1, 5, 4, 5, 5, 4, 4, 1, 1, 1, 6, 5, 7, 5, 7, 4, 5, 1, 1, 1, 7, 6, 9, 7, 7, 7, 5, 5, 1, 1, 1, 9, 7, 11, 10, 10, 7, 9, 5, 6, 1, 1, 1, 10, 9, 13, 12, 14, 10, 9, 9, 6, 6, 1, 1, 1, 12, 10, 17, 15, 17, 15

Offset: 1

Emeric Deutsch, Feb 12 2006

Row sums yield A000041. Column 0 is A003114. Sum_{k=0..n-1} k*T(n,k) = A116365(n).

			T(6,2) = 3 because we have [4,1,1], [2,2,2] and [2,2,1,1] (the bottom tails are [1,1], [2] and [1,1], respectively, each being a partition of 2).

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).

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A000041, A003114, A115994, A115995, A116365, A114088, A114089.

g:=sum(z^(k^2)/product((1-z^j),j=1..k)/product((1-(t*z)^i),i=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
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
T:= (n, k)-> add(b(k, d)*b(n-d^2-k, d), d=0..floor(sqrt(n))):
seq(seq(T(n, k), k=0..n-1), n=1..20); # Alois P. Heinz, Apr 09 2012

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]] ]; T[n_, k_] := Sum[b[k, d]*b[n-d^2-k, d], {d, 0, Floor[Sqrt[n]]}]; Table[Table[ T[n, k], {k, 0, n-1}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

G.f.: Sum_(q^(k^2)/Product_((1-q^j)(1-(t*q)^j), j=1..k), k=1..infinity).

A331553 Irregular triangle T(n,k) = A115722(n,k)^2 - n.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 3, 3, 0, 3, 3, 4, 4, 1, 4, 1, 4, 4, 5, 5, 2, 5, 2, 2, 5, 2, 2, 5, 5, 6, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 3, 3, 6, 6, 7, 7, 4, 7, 4, 4, 7, 4, 4, 4, 4, 7, 4, 4, 4, 4, 7, 4, 4, 4, 7, 7, 8, 8, 5, 8, 5, 5, 8, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 0, 5

Offset: 0

Michael De Vlieger, Jan 20 2020

Let P be an integer partition of n, and let D be the Durfee square of P with side length s, thus area s^2. We borrow the term "square excess" from A053186(n), which is simply the difference n - floor(sqrt(n)). This sequence lists the "Durfee square excess" of P = n - s^2 for all partitions P of n in reverse lexicographic order.
Zero appears in row n for n that are perfect squares. Let r = sqrt(n). For perfect square n, there exists a partition of n that consists of a run of r parts that are each r themselves; e.g., for n = 4, we have {2, 2}, for n = 9, we have {3, 3, 3}. It is clear through the Ferrers diagram of these partitions that they are equivalent to their Durfee square, thus n - s^2 = 0.
Since the partitions of any n contain Durfee squares in the range of 1 <= s <= floor(sqrt(n)) (with perfect square n also including k = 0), the distinct Durfee square excesses must be the differences n - s^2 for 1 <= s <= floor(sqrt(n)).

			Table begins:
0: 0;
1: 0;
2: 1, 1;
3: 2, 2, 2;
4: 3, 3, 0, 3, 3;
5: 4, 4, 1, 4, 1, 4, 4;
6: 5, 5, 2, 5, 2, 2, 5, 2, 2, 5, 5;
7: 6, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 3, 3, 6, 6;
...
Table of distinct terms:
1:  0;
2:  1;
3:  2;
4:  0,  3;
5:  1,  4;
6:  2,  5;
7:  3,  6;
8:  4,  7;
9:  0,  5,  8;
...
For n = 4, the partitions are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}. The partition {2, 2} has Durfee square s = 2; for all partitions except {2, 2}, we have Durfee square with s = 1.

Eric Weisstein's World of Mathematics, Durfee Square.

Cf. A115722, A116365.

{0}~Join~Array[Map[Total@ # - Block[{k = Length@ #}, While[Nand[k > 0, AllTrue[Take[#, k], # >= k &]], k--]; k]^2 &, IntegerPartitions[#]] &, 12] // Flatten

T(n,k) = A115722(n,k)^2 - n.

2 * A116365(n) = sum of row n.

cp's OEIS Frontend

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).

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A114089 Total number of parts in the tails below the Durfee squares of all partitions of n.

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A114087 Triangle read by rows: T(n,k) is the number of partitions of n whose tails below their Durfee squares have size k (n>=1; 0<=k<=n-1).

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Maple

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A331553 Irregular triangle T(n,k) = A115722(n,k)^2 - n.

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