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%I #66 Nov 27 2020 23:43:07
%S 1,1,1,1,1,2,1,1,1,1,2,3,3,3,3,2,1,1,1,1,2,3,5,5,7,7,8,7,7,5,5,3,2,1,
%T 1,1,1,2,3,5,7,9,11,14,16,18,19,20,20,19,18,16,14,11,9,7,5,3,2,1,1,1,
%U 1,2,3,5,7,11,13,18,22,28,32,39,42,48,51,55,55,58,55,55,51,48,42,39,32,28
%N Triangle read by rows giving number of partitions of k (k=0 .. n^2) with Ferrers plot fitting in an n X n box.
%C Seems to approximate a Gaussian distribution, the sum of all 1+n^2 terms in a row equals the central binomial coefficients.
%C a(n,k) is the number of sequences of n 0's and n 1's having major index equal to k (the major index is the sum of the positions of the 1's that are immediately followed by 0's). Equivalently, a(n,k) is the number of Grand Dyck paths of length 2n for which the sum of the positions of the valleys is k. Example: a(3,7)=2 because the only sequences of three 0's and three 1's with major index 7 are 010110 and 110010. The corresponding Grand Dyck paths are obtained by replacing a 0 by a U=(1,1) step and a 1 by a D=(1,-1) step. - _Emeric Deutsch_, Oct 02 2007
%C Also, number of n-multisets in [0..n] whose elements sum up to n. - _M. F. Hasler_, Apr 12 2012
%C Let P be the poset [n] X [n] ordered by the product order. Let J(P) be the set of all order ideals of P, ordered by inclusion. Then J(P) is a finite sublattice of Young's lattice and T(n,k) is the number of elements in J(P) that have rank k. - _Geoffrey Critzer_, Mar 26 2020
%D G. E. Andrews and K. Eriksson, Integer partitions, Cambridge Univ. Press, 2004, pp. 67-69.
%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; exercise 3.2.3.
%D A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.
%H Alois P. Heinz, <a href="/A063746/b063746.txt">Rows k = 0..31, flattened</a>
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, 2009, page 45.
%H A. V. Yurkin, <a href="http://www.mce.biophys.msu.ru/eng/archive/abstracts/mce19/sect1138/doc150220/">On similarity of systems of geometrical and arithmetic triangles</a>, in Mathematics, Computing, Education Conference XIX, 2012.
%H A. V. Yurkin, <a href="http://arxiv.org/abs/1302.6287">New view on the diffraction discovered by Grimaldi and Gaussian beams</a>, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.
%F Table[T[k, n, n], {n, 0, 9}, {k, 0, n^2}] with T[ ] defined as in A047993.
%F G.f.: Consider a function; f(n) = 1 + sum(i_1=1, n, sum(i_2=0, i_1, ..., sum(i_n=0, i_(n-1), x^(sum(j=1, n, i_j))*(1+...+x^i_n))...)) Then the GF is f(1)+x^3.f(2)+x^8.f(3)+..., where after x^3 the increase is n^2+1 from f(n). - _Jon Perry_, Jul 13 2004
%F G.f. for n-th row is obtained if we set x(i) = 1+x^i+x^(2*i)+...+x^(n*i), i=1, 2, ..., n, in the cycle index Z(S(n);x(1), x(2), ..., x(n)) of the symmetric group S(n) of degree n. - _Vladeta Jovovic_, Dec 17 2004
%F G.f. of row n: the q-binomial coefficient [2n,n]. - _Emeric Deutsch_, Apr 23 2007
%F T(n,k)=1 for k=0,1,n^2-1,n^2. For all m>n, T(m,n)=T(n,n)=A000041(n), i.e., below the diagonal the columns remain constant, because there cannot be more than n nonzero elements with sum <= n. - _M. F. Hasler_, Apr 12 2012
%F T(n,2n) = A128552(n-2). - _Geoffrey Critzer_, Sep 27 2013
%e From _M. F. Hasler_, Apr 12 2012: (Start)
%e The table reads:
%e n=0: 1 _ (k=0)
%e n=1: 1 1 _ (k=0..1)
%e n=2: 1 1 2 1 1 _ (k=0..4)
%e n=3: 1 1 2 3 3 3 3 2 1 1 _ (k=0..9)
%e n=4: 1 1 2 3 5 5 7 7 8 7 7 5 5 3 2 1 1 _ (k=0..16)
%e n=5: 1 1 2 3 5 7 9 11 14 16 18 19 20 20 19 18 16 ... _ (k=0..25)
%e etc. (End)
%e Cycle index of S(3) is (1/6)*(x(1)^3+3*x(1)*x(2)+2*x(3)), so g.f. for 3rd row is (1/6)*((1+x+x^2+x^3)^3+3*(1+x+x^2+x^3)*(1+x^2+x^4+x^6)+2*(1+x^3+x^6+x^9) = x^9+x^8+2*x^7+3*x^6+3*x^5+3*x^4+3*x^3+2*x^2+x+1.
%e a(3,7)=2 because the only partitions of 7 with Ferrers plot fitting into a 3 X 3 box are [3,3,1] and [3,2,2].
%p for n from 0 to 15 do QBR[n]:=sum(q^i,i=0..n-1) od: for n from 0 to 15 do QFAC[n]:=product(QBR[j],j=1..n) od: qbin:=(n,k)->QFAC[n]/QFAC[k]/QFAC[n-k]: for n from 0 to 7 do P[n]:=sort(expand(simplify(qbin(2*n,n)))) od: for n from 0 to 7 do seq(coeff(P[n],q,j),j=0..n^2) od; # yields sequence in triangular form - _Emeric Deutsch_, Apr 23 2007
%p # second Maple program:
%p b:= proc(n, i, k) option remember;
%p `if`(n=0, 1, `if`(i<1 or k<1, 0, b(n, i-1, k)+
%p `if`(i>n, 0, b(n-i, i, k-1))))
%p end:
%p T:= n-> seq(b(k, min(n, k), n), k=0..n^2):
%p seq(T(n), n=0..8); # _Alois P. Heinz_, Apr 05 2012
%t Table[nn=n^2;CoefficientList[Series[Product[(1-x^(n+i))/(1-x^i),{i,1,n}],{x,0,nn}],x],{n,0,6}]//Grid (* _Geoffrey Critzer_, Sep 27 2013 *)
%t Table[CoefficientList[QBinomial[2n,n,q] // FunctionExpand, q], {n,0,6}] // Flatten (* _Peter Luschny_, Jul 22 2016 *)
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1 || k < 1, 0, b[n, i - 1, k] + If[i > n, 0, b[n - i, i, k - 1]]]];
%t T[n_] := Table[b[k, Min[n, k], n], {k, 0, n^2}];
%t Table[T[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)
%o (PARI) f1(x)=1+x*sum(j=0,1,x^j); f2(x)=1+sum(i=1,2,x^i*sum(j=0,i,x^j)); f3(x)=1+sum(i=1,3,sum(k=0,i,x^(i+k)*sum(j=0,k,x^j))); f4(x)=1+sum(i=1,4,sum(i1=0,i,sum(k=0,i1,x^(i+i1+k)*sum(j=0,k,x^j)))) f(x)=f1(x)+x^3*f2(x)+x^8*f3(x)+x^18*f4(x); for (i=0,30,print1(","polcoeff(f(x),i))) (Perry)
%o (PARI) T(n,k)=polcoeff(prod(i=0,n,sum(j=0,n,x^(j*i*(n^2+n+1)+j),O(x^(k*(n^2+n+1)+n+1)))),k*(n^2+n+1)+n) /* Based on a more general formula due to R. Gerbicz */ _M. F. Hasler_, Apr 12 2012
%Y Cf. A008968, A047971, A047993.
%Y Row lengths are given by A002522. - _M. F. Hasler_, Apr 14 2012
%Y Antidiagonal sums are given by A260894.
%K nonn,tabf
%O 0,6
%A _Wouter Meeussen_, Aug 14 2001
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