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%I A198890 #42 Aug 23 2023 08:43:02
%S A198890 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,
%T A198890 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,
%U A198890 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
%N 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.
%H A198890 Peter Bala, <a href="/A198890/a198890.pdf">A note on A198890</a>
%H A198890 J. Striker, <a href="https://arxiv.org/abs/1002.3391">A direct bijection between descending plane partitions with no special parts and permutation matrices</a>, arXiv:1002.3391 [math.CO], 2010-2012.
%H A198890 J. Striker, <a href="https://doi.org/10.1016/j.disc.2011.07.030">A direct bijection between descending plane partitions with no special parts and permutation matrices</a>, Discrete Math., 311 (2011).
%F A198890 From _Peter Bala_, May 29 2022: (Start)
%F A198890 T(0, 0) = 1; T(1, 0) = 1.
%F A198890 T(n, k) = 0 for k < 0 or k > (1/3)*(n+1)*n*(n-1).
%F A198890 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).
%F A198890 T(n,k) = T(n, (1/3)*(n+1)*n*(n-1) - k).
%F A198890 Sum_{k = 0..(1/3)*(n+1)*n*(n-1)} T(n, k) = n!.
%F A198890 Sum_{k = 0..(1/3)*(n+1)*n*(n-1)} (-1)^k*T(n, k) = A037223(n).
%F A198890 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.
%F A198890 n-th row polynomial R(n,x) = Product_{j = 1..n} (1 - x^(j^2))/(1 - x^j).
%F A198890 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.
%F A198890 The x-adic limit of R(n,x) as n -> 00 is the g.f. of A087153. (End)
%e A198890 Rows 1 through 5 are
%e A198890 1
%e A198890 1, 0, 1
%e A198890 1, 0, 1, 1, 0, 1, 1, 0, 1
%e A198890 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1
%e A198890 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
%e A198890 From _Peter Bala_, May 29 2022: (Start)
%e A198890 Row 3 generating polynomial:
%e A198890 Permutation p Pairs (p(i),p(j)) with p(i) > p(j) inv_1(p)
%e A198890 123 - 0
%e A198890 132 (3,2) 3
%e A198890 213 (2,1) 2
%e A198890 231 (2,1), (3,1) 5
%e A198890 312 (3,1), (3,2) 6
%e A198890 321 (3,2), (3,1), (2,1) 8
%e A198890 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)
%p A198890 s:=(k,q)->add(q^i,i=0..k-1);
%p A198890 f:=n->mul(s(i,q^i),i=1..n);
%p A198890 g:=n->seriestolist(series(f(n),q,1000));
%p A198890 for n from 1 to 10 do lprint(g(n)); od:
%p A198890 # alternative program
%p A198890 T := proc (n, k) option remember;
%p A198890 if n = 0 or n = 1 and k = 0 then 1
%p A198890 elif k > ((1/3)*n-1/3)*n*(n+1) then 0
%p A198890 elif k < 0 then 0
%p A198890 else T(n, k-n) + T(n-1, k) - T(n-1, k-n^2) fi end:
%p A198890 seq(print(seq(T(n, k), k = 0..(1/3)*(n-1)*n*(n+1))), n = 1..6); # _Peter Bala_, Jun 07 2022
%Y A198890 Row sums give A000142 (factorial numbers).
%Y A198890 Cf. A001754, A008302, A037223, A087153.
%K A198890 nonn,tabf
%O A198890 1,20
%A A198890 _N. J. A. Sloane_, Oct 30 2011
%E A198890 Name clarified by _Ludovic Schwob_, Jun 15 2023