

A063746


Triangle read by rows giving number of partitions of k (k=0 .. n^2) with Ferrers plot fitting in an n X n box.


11



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, 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, 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
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OFFSET

0,6


COMMENTS

Seems to approximate a Gaussian distribution, the sum of all 1+n^2 terms in a row equals the central binomial coefficients.
a(n,k) is the number of sequences of n 0s and n 1s having major index equal to k (the major index is the sum of the positions of the 1s that are immediately followed by 0s). 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 3 0s and 3 1s 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
Also, number of nmultisets in [0..n] whose elements sum up to n.  M. F. Hasler, Apr 12 2012
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


REFERENCES

G. E. Andrews and K. Eriksson, Integer partitions, Cambridge Univ. Press, 2004, pp. 6769.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; exercise 3.2.3.
A. V. Yurkin, New binomial and new view on light theory, (book), 2013, 78 pages, no publisher listed.


LINKS

Alois P. Heinz, Rows k = 0..31, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 45.
A. V. Yurkin, On similarity of systems of geometrical and arithmetic triangles, in Mathematics, Computing, Education Conference XIX, 2012.
A. V. Yurkin, New view on the diffraction discovered by Grimaldi and Gaussian beams, arXiv preprint arXiv:1302.6287 [physics.optics], 2013.


FORMULA

Table[T[k, n, n], {n, 0, 9}, {k, 0, n^2}] with T[ ] defined as in A047993.
G.f.: Consider a function; f(n) = 1 + sum(i_1=1, n, sum(i_2=0, i_1, ..., sum(i_n=0, i_(n1), 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
G.f. for nth 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
G.f. of row n: the qbinomial coefficient [2n,n].  Emeric Deutsch, Apr 23 2007
T(n,k)=1 for k=0,1,n^21,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
T(n,2n) = A128552(n2).  Geoffrey Critzer, Sep 27 2013


EXAMPLE

From M. F. Hasler, Apr 12 2012: (Start)
The table reads:
n=0: 1 _ (k=0)
n=1: 1 1 _ (k=0..1)
n=2: 1 1 2 1 1 _ (k=0..4)
n=3: 1 1 2 3 3 3 3 2 1 1 _ (k=0..9)
n=4: 1 1 2 3 5 5 7 7 8 7 7 5 5 3 2 1 1 _ (k=0..16)
n=5: 1 1 2 3 5 7 9 11 14 16 18 19 20 20 19 18 16 ... _ (k=0..25)
etc. (End)
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.
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].


MAPLE

for n from 0 to 15 do QBR[n]:=sum(q^i, i=0..n1) 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[nk]: 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
# second Maple program:
b:= proc(n, i, k) option remember;
`if`(n=0, 1, `if`(i<1 or k<1, 0, b(n, i1, k)+
`if`(i>n, 0, b(ni, i, k1))))
end:
T:= n> seq(b(k, min(n, k), n), k=0..n^2):
seq(T(n), n=0..8); # Alois P. Heinz, Apr 05 2012


MATHEMATICA

Table[nn=n^2; CoefficientList[Series[Product[(1x^(n+i))/(1x^i), {i, 1, n}], {x, 0, nn}], x], {n, 0, 6}]//Grid (* Geoffrey Critzer, Sep 27 2013 *)
Table[CoefficientList[QBinomial[2n, n, q] // FunctionExpand, q], {n, 0, 6}] // Flatten (* Peter Luschny, Jul 22 2016 *)


PROG

(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)
(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


CROSSREFS

Cf. A008968, A047971, A047993.
Row lengths are given by A002522.  M. F. Hasler, Apr 14 2012
Antidiagonal sums are given by A260894.
Sequence in context: A247378 A094102 A220091 * A293429 A201075 A131338
Adjacent sequences: A063743 A063744 A063745 * A063747 A063748 A063749


KEYWORD

nonn,tabf,changed


AUTHOR

Wouter Meeussen, Aug 14 2001


STATUS

approved



