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 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. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,20 LINKS Table of n, a(n) for n=1..146. 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). 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) EXAMPLE 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) 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 CROSSREFS Row sums give A000142 (factorial numbers). Cf. A001754, A008302, A037223, A087153. Sequence in context: A140195 A196564 A196563 * A305831 A022927 A063435 Adjacent sequences: A198887 A198888 A198889 * A198891 A198892 A198893 KEYWORD nonn,tabf AUTHOR N. J. A. Sloane, Oct 30 2011 EXTENSIONS Name clarified by Ludovic Schwob, Jun 15 2023 STATUS approved

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Last modified December 4 11:15 EST 2023. Contains 367560 sequences. (Running on oeis4.)