A237770 - OEIS

%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