A198890 Irregular triangle read by rows: row n gives expansion of g.f. for descending plane partitions of order n with no special parts and weight equal to sum of the parts.

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 4, 5, 5, 4, 6, 4, 5, 5, 4, 5, 4, 4, 4, 3, 4, 2, 3, 2, 2, 2, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 3, 2, 4, 3, 5, 5, 7, 6, 8, 8, 9, 10, 12, 10, 14, 12, 14, 15, 16, 15, 18, 16, 18, 18, 20, 17, 21, 18, 20, 20, 20, 18, 21, 17, 20, 18, 18, 16, 18, 15, 16, 15, 14, 12, 14, 10, 12, 10, 9, 8, 8, 6, 7, 5, 5, 3, 4, 2, 3, 2, 1, 1, 1, 0, 1

Offset: 1

Author

N. J. A. Sloane, Oct 30 2011

Keywords

• nonn
• tabf

Examples

			Rows 1 through 5 are
  1
  1, 0, 1
  1, 0, 1, 1, 0, 1, 1, 0, 1
  1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1
  1, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 4, 5, 5, 4, 6, 4, 5, 5, 4, 5, 4, 4, 4, 3, 4, 2, 3, 2, 2, 2, 1, 1, 1, 0, 1
From _Peter Bala_, May 29 2022: (Start)
Row 3 generating polynomial:
   Permutation p    Pairs (p(i),p(j)) with p(i) > p(j)       inv_1(p)
       123                       -                              0
       132                     (3,2)                            3
       213                     (2,1)                            2
       231                 (2,1), (3,1)                         5
       312                 (3,1), (3,2)                         6
       321              (3,2), (3,1), (2,1)                     8
Hence R(3,x) = x^0 + x^2 + x^3 + x^5 + x^6 + x^8 = (1 + x^2)*(1 + x^3 + x^6) = ((1 - x^4)/(1 - x^2)) * (1 - x^9)/(1 - x^3). (End)
		

Links

• Peter Bala, A note on A198890
• J. Striker, A direct bijection between descending plane partitions with no special parts and permutation matrices, arXiv:1002.3391 [math.CO], 2010-2012.
• J. Striker, A direct bijection between descending plane partitions with no special parts and permutation matrices, Discrete Math., 311 (2011).

Crossrefs

Row sums give A000142 (factorial numbers).

Cf. A001754, A008302, A037223, A087153.

Programs

Maple

s:=(k,q)->add(q^i,i=0..k-1);
f:=n->mul(s(i,q^i),i=1..n);
g:=n->seriestolist(series(f(n),q,1000));
for n from 1 to 10 do lprint(g(n)); od:
# alternative program
T := proc (n, k) option remember;
if n = 0 or n = 1 and k = 0 then 1
elif k > ((1/3)*n-1/3)*n*(n+1) then 0
elif k < 0 then 0
else T(n, k-n) + T(n-1, k) - T(n-1, k-n^2) fi end:
seq(print(seq(T(n, k), k = 0..(1/3)*(n-1)*n*(n+1))), n = 1..6); # Peter Bala, Jun 07 2022

Formula

From Peter Bala, May 29 2022: (Start)

T(0, 0) = 1; T(1, 0) = 1.

T(n, k) = 0 for k < 0 or k > (1/3)*(n+1)*n*(n-1).

T(n, k) = Sum_{j = 0..n-1} T(n-1, k-n*j); T(n, k) = T(n, k-n) + T(n-1, k) - T(n-1, k-n^2).

T(n,k) = T(n, (1/3)*(n+1)*n*(n-1) - k).

Sum_{k = 0..(1/3)*(n+1)*n*(n-1)} T(n, k) = n!.

Sum_{k = 0..(1/3)*(n+1)*n*(n-1)} (-1)^k*T(n, k) = A037223(n).

Sum_{k = 0..(1/3)*(n+1)*n*(n-1)} k*T(n, k) = (1/3)*n!*binomial(n-1,2) = 2*A001754(n) for n >= 1.

n-th row polynomial R(n,x) = Product_{j = 1..n} (1 - x^(j^2))/(1 - x^j).

let k be a nonnegative integer. Let p = p(1)p(2)...p(n) be a permutation of {1,2,...,n}. We define the k-th inversion number of p by inv_k(p) = Sum_{pairs (i,j), 1 <= i < j <= n, such that p(i) > p(j)} (p(i))^k. The n-th row polynomial R(n,x) equals Sum_{permutations p of {1,2,...,n} } x^(inv_1(p)). An example is given below. For the case k = 0 see A008302.

The x-adic limit of R(n,x) as n -> 00 is the g.f. of A087153. (End)

Extensions

Name clarified by Ludovic Schwob, Jun 15 2023