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%I #55 Jan 22 2020 14:37:20
%S 1,1,1,2,4,9,22,59,170,516,1658,5583,19683,72162,274796,1082439,
%T 4406706,18484332,79818616,353995743,1611041726,7510754022,
%U 35842380314,174850257639,871343536591,4430997592209,22978251206350,121410382810005,653225968918521
%N Number of standard Young tableaux with n cells without a succession v, v+1 in a row.
%C A standard Young tableau (SYT) without a succession v, v+1 in a row is called a nonconsecutive tableau.
%C Also the number of ballot sequences without two consecutive elements equal. A ballot sequence B is a string such that, for all prefixes P of B, h(i)>=h(j) for i<j, where h(x) is the number of times x appears in P (see A000085).
%C First column (k=0) of A238125.
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A237770/b237770.txt">Table of n, a(n) for n = 0..68</a> (terms 0..48 from Alois P. Heinz)
%H Timothy Y. Chow, Henrik Eriksson and C. Kenneth Fan, <a href="https://doi.org/10.37236/1895">Chess Tableaux</a>, The Electronic Journal of Combinatorics, vol.11, no.2, (2005).
%H S. Dulucq and O. Guibert, <a href="https://doi.org/10.1016/S0012-365X(96)83009-3">Stack words, standard tableaux and Baxter permutations</a>, Disc. Math. 157 (1996), 91-106.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F a(n) = Sum_{k=1..A264078(n)} k * A264051(n,k). - _Alois P. Heinz_, Nov 02 2015
%e The a(5) = 9 such tableaux of 5 are:
%e [1] [2] [3] [4] [5] [6] [7] [8] [9]
%e 135 13 135 13 13 14 14 15 1
%e 24 24 2 25 2 25 2 2 2
%e 5 4 4 4 3 3 3 3
%e 5 5 4 4
%e 5
%e The corresponding ballot sequences are:
%e 1: [ 0 1 0 1 0 ]
%e 2: [ 0 1 0 1 2 ]
%e 3: [ 0 1 0 2 0 ]
%e 4: [ 0 1 0 2 1 ]
%e 5: [ 0 1 0 2 3 ]
%e 6: [ 0 1 2 0 1 ]
%e 7: [ 0 1 2 0 3 ]
%e 8: [ 0 1 2 3 0 ]
%e 9: [ 0 1 2 3 4 ]
%p h:= proc(l, j) option remember; `if`(l=[], 1,
%p `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
%p `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
%p h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
%p end:
%p g:= proc(n, i, l) `if`(n=0 or i=1, h([1$n, l[]], 0),
%p `if`(i<1, 0, g(n, i-1, l)+
%p `if`(i>n, 0, g(n-i, i, [i, l[]]))))
%p end:
%p a:= n-> g(n, n, []):
%p seq(a(n), n=0..30);
%p # second Maple program (counting ballot sequences):
%p b:= proc(n, v, l) option remember;
%p `if`(n<1, 1, add(`if`(i<>v and (i=1 or l[i-1]>l[i]),
%p b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
%p b(n-1, nops(l)+1, [l[], 1]))
%p end:
%p a:= proc(n) option remember; forget(b); b(n-1, 1, [1]) end:
%p seq(a(n), n=0..30);
%t b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Sum[If[i != v && (i == 1 || l[[i-1]] > l[[i]]), b[n-1, i, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; a[n_] := a[n] = b[n-1, 1, {1}]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 06 2015, translated from 2nd Maple program *)
%Y Cf. A000085 (all Young tableaux), A000957, A001181, A214021, A214087, A214159, A214875.
%Y Cf. A238126 (tableaux with one succession), A238127 (two successions).
%Y Cf. A264051, A264078.
%K nonn
%O 0,4
%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 13 2014
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