|
%I #43 Jun 19 2018 05:56:45
%S 1,0,1,0,1,1,0,3,0,1,0,7,2,0,1,0,20,5,0,0,1,0,56,14,5,0,0,1,0,182,35,
%T 14,0,0,0,1,0,589,132,28,14,0,0,0,1,0,2088,399,90,42,0,0,0,0,1,0,7522,
%U 1556,285,90,42,0,0,0,0,1,0,28820,5346,1232,165,132,0,0,0,0,0,1
%N Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k largest parts, n >= k >= 0.
%C Also number of standard Young tableaux with last row of length k.
%H Joerg Arndt and Alois P. Heinz, <a href="/A238123/b238123.txt">Rows n = 0..60, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e Triangle starts:
%e 00: 1;
%e 01: 0, 1;
%e 02: 0, 1, 1;
%e 03, 0, 3, 0, 1;
%e 04: 0, 7, 2, 0, 1;
%e 05: 0, 20, 5, 0, 0, 1;
%e 06: 0, 56, 14, 5, 0, 0, 1;
%e 07: 0, 182, 35, 14, 0, 0, 0, 1;
%e 08: 0, 589, 132, 28, 14, 0, 0, 0, 1;
%e 09: 0, 2088, 399, 90, 42, 0, 0, 0, 0, 1;
%e 10: 0, 7522, 1556, 285, 90, 42, 0, 0, 0, 0, 1;
%e 11: 0, 28820, 5346, 1232, 165, 132, 0, 0, 0, 0, 0, 1;
%e 12: 0, 113092, 21515, 4378, 737, 297, 132, 0, 0, 0, 0, 0, 1;
%e 13: 0, 464477, 82940, 17082, 3003, 572, 429, 0, 0, 0, 0, 0, 0, 1;
%e ...
%e The T(6,2)=14 ballot sequences of length 6 with 2 maximal elements are (dots for zeros):
%e 01: [ . . . . 1 1 ]
%e 02: [ . . . 1 . 1 ]
%e 03: [ . . . 1 1 . ]
%e 04: [ . . 1 . . 1 ]
%e 05: [ . . 1 . 1 . ]
%e 06: [ . . 1 1 . . ]
%e 07: [ . . 1 1 2 2 ]
%e 08: [ . . 1 2 1 2 ]
%e 09: [ . 1 . . . 1 ]
%e 10: [ . 1 . . 1 . ]
%e 11: [ . 1 . 1 . . ]
%e 12: [ . 1 . 1 2 2 ]
%e 13: [ . 1 . 2 1 2 ]
%e 14: [ . 1 2 . 1 2 ]
%e The T(8,4)=14 such ballot sequences of length 8 and 4 maximal elements are:
%e 01: [ . . . . 1 1 1 1 ]
%e 02: [ . . . 1 . 1 1 1 ]
%e 03: [ . . . 1 1 . 1 1 ]
%e 04: [ . . . 1 1 1 . 1 ]
%e 05: [ . . 1 . . 1 1 1 ]
%e 06: [ . . 1 . 1 . 1 1 ]
%e 07: [ . . 1 . 1 1 . 1 ]
%e 08: [ . . 1 1 . . 1 1 ]
%e 09: [ . . 1 1 . 1 . 1 ]
%e 10: [ . 1 . . . 1 1 1 ]
%e 11: [ . 1 . . 1 . 1 1 ]
%e 12: [ . 1 . . 1 1 . 1 ]
%e 13: [ . 1 . 1 . . 1 1 ]
%e 14: [ . 1 . 1 . 1 . 1 ]
%e These are the (reversed) Dyck words of semi-length 4.
%p b:= proc(n, l) option remember; `if`(n<1, x^l[-1],
%p b(n-1, [l[], 1]) +add(`if`(i=1 or l[i-1]>l[i],
%p b(n-1, subsop(i=l[i]+1, l)), 0), i=1..nops(l)))
%p end:
%p T:= n->`if`(n=0, 1, (p->seq(coeff(p, x, i), i=0..n))(b(n-1, [1]))):
%p seq(T(n), n=0..12);
%p # second Maple program (counting SYT):
%p h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p end:
%p g:= proc(n, i, l) `if`(n=0 or i=1, h([l[], 1$n])*x^`if`(n>0, 1,
%p `if`(l=[], 0, l[-1])), g(n, i-1, l)+
%p `if`(i>n, 0, g(n-i, i, [l[], i])))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(g(n, n, [])):
%p seq(T(n), n=0..12);
%t b[n_, l_List] := b[n, l] = If[n<1, x^l[[-1]], b[n-1, Append[l, 1]] + Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; T[n_] := If[n == 0, 1, Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, {1}]]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 07 2015, translated from Maple *)
%o (PARI) (A238123(n,k)=if(k, vecsum(apply(p->n!/Hook(Vecrev(p)), select(p->p[1]==k,partitions(n,[k,n])))), !n)); Hook(P,h=vector(P[1]),L=P[#P])={prod(i=1, L, h[i]=L-i+1)*prod(i=1,#P-1, my(D=-L+L=P[#P-i]); prod(k=0,L-1,h[L-k]+=min(k,D)+1))} \\ _M. F. Hasler_, Jun 03 2018
%Y The terms T(2*n,n) are the Catalan numbers (A000108).
%Y Columns k=0-10 give: A000007, A238124, A244099, A244100, A244101, A244102, A244103, A244104, A244105, A244106, A244107.
%Y Row sums give A000085.
%Y Cf. A026794.
%K nonn,tabl
%O 0,8
%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 21 2014
|
