A388724 Triangle read by rows where T(n,k) is the number of strictly superdiagonal reversed integer partitions of n with maximum k.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 2, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 2, 2, 1, 1, 1, 0, 1

Offset

0,60

Comments

A sequence (y_1, ..., y_k) is strictly superdiagonal iff y_i > i for all i = 1..k.

Links

John Tyler Rascoe, Rows n = 0..100, flattened

Formula

Conjecture: Column sums are A000245.

G.f.: A(q,t) = Sum_{m>0} P(m,q,t) where P(m,q,t) = 1 for m =< 0, P(m,q,t) = Pr(m,q,t) * t * q^m/(1 - t*q^m) for m > 0, Pr(m,q,t) = Sum_{j=1..m} q^((j/2)*(-j+2*m+3)-1) * t^j * Ca(j-1,q) * P(m-j,q,t), and Ca(n,q) is the n-th row polynomial of A227543. - John Tyler Rascoe, Oct 04 2025

Example

Row n = 16 counts the following partitions for k = 5..9:
  (2,4,5,5)  (4,6,6)    (2,7,7)    (8,8)    (7,9)
  (3,3,5,5)  (5,5,6)    (3,6,7)    (2,6,8)  (2,5,9)
  (3,4,4,5)  (2,3,5,6)  (4,5,7)    (3,5,8)  (3,4,9)
             (2,4,4,6)  (2,3,4,7)  (4,4,8)
             (3,3,4,6)
Triangle begins:
  0
  0  0
  0  0  1
  0  0  0  1
  0  0  0  0  1
  0  0  0  1  0  1
  0  0  0  1  1  0  1
  0  0  0  0  1  1  0  1
  0  0  0  0  1  1  1  0  1
  0  0  0  0  1  1  1  1  0  1
  0  0  0  0  2  2  1  1  1  0  1
  0  0  0  0  1  2  2  1  1  1  0  1
  0  0  0  0  1  2  3  2  1  1  1  0  1
  0  0  0  0  0  2  2  3  2  1  1  1  0  1
  0  0  0  0  0  2  3  3  3  2  1  1  1  0  1
  0  0  0  0  0  4  3  3  3  3  2  1  1  1  0  1
  0  0  0  0  0  3  5  4  4  3  3  2  1  1  1  0  1
  0  0  0  0  0  3  5  6  4  4  3  3  2  1  1  1  0  1
  0  0  0  0  0  2  6  6  7  5  4  3  3  2  1  1  1  0  1
  0  0  0  0  0  1  4  8  7  7  5  4  3  3  2  1  1  1  0  1
  0  0  0  0  0  1  5  7  9  8  8  5  4  3  3  2  1  1  1  0  1

Mathematica

suppstrQ[mset_]:=And@@Table[mset[[i]]>i, {i, Length[mset]}];
Table[Length[Select[Reverse/@IntegerPartitions[n], Max@@#==k&&suppstrQ[#]&]], {n, 0, 15}, {k, 0, n}]

Prog

(PARI)
tri(n) = {(n*(n+1)/2)}
b(n) = {floor(1/2+sqrt(2*n))}
C(q, n) = {c = if(n<=1, return(1)); return(sum(i=0, n-1, C(q, i) * C(q, n-1-i) * q^((i+1)*(n-1 -i)))); }
Ca(n) = {polrecip(C(q, n))}
A388724_q(rowmax) = {my(N = rowmax+1, pl= vector(b(N), i, 1)); for(m=1, b(N), my(g = sum(j=1, m, q^(tri(j)+j*(m-j)+j-1) * t^j *Ca(j-1) * if(j<m, pl[m-j]/(1-t*q^(m-j)), 1))); pl[m]=g); my(f = sum(m=1, b(N), t*q^m* pl[m]/(1-t*q^m))  + O('q^(N+1))); vector(N, n, if(n==2, [0, 0], Vecrev(polcoeff(f, n-1))))} \\ John Tyler Rascoe, Oct 04 2025

Crossrefs

Row sums are A238394 (complement A238395), non-strictly A238873.

The subdiagonal version is A387880.

The non-strictly version is A388713.

A000041 counts integer partitions, strict A000009.

A001522 counts partitions with a diagonal, complement A064428.

A003106 counts strictly superdiagonal partitions, strict A237979, ranks A352830.

A003114 counts superdiagonal partitions, complement A387118, strict partitions A025157.

A114088 counts partitions by excedances.

A115720 and A115994 count partitions by their Durfee square.

A352833 counts partitions by fixed points, reversed partitions A238352.

A370592 counts choosable partitions (ranks A368100), complement A370593 (ranks A355529).

Cf. A000700, A052335, A052337, A227543, A238875, A352828, A370585, A388710, A388711.

Sequence in context: A263087 A204433 A004578 * A389627 A319372 A319367

Adjacent sequences: A388721 A388722 A388723 * A388725 A388726 A388727

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 24 2025

Status

approved