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A238128 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having largest descent k, n>=0, 0<=k<=n. 12
1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 4, 1, 0, 0, 7, 13, 5, 1, 0, 0, 11, 37, 21, 6, 1, 0, 0, 15, 100, 78, 31, 7, 1, 0, 0, 22, 265, 292, 133, 43, 8, 1, 0, 0, 30, 694, 1028, 586, 215, 57, 9, 1, 0, 0, 42, 1828, 3691, 2453, 1073, 325, 73, 10, 1, 0, 0, 56, 4815, 13004, 10357, 5058, 1836, 467, 91, 11, 1, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Also number of standard Young tableaux with a pair of cells (v,v+1) such that v lies k rows above v+1, and no pair (u,u+1) with a larger such separation exists.

LINKS

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..35, flattened

EXAMPLE

Triangle starts:

00:  1;

01:  1,     0;

02:  2,     0,     0;

03:  3,     1,     0,     0;

04:  5,     4,     1,     0,     0;

05:  7,    13,     5,     1,     0,    0;

06: 11,    37,    21,     6,     1,    0,    0;

07: 15,   100,    78,    31,     7,    1,    0,   0;

08: 22,   265,   292,   133,    43,    8,    1,   0,   0;

09: 30,   694,  1028,   586,   215,   57,    9,   1,   0,  0;

10: 42,  1828,  3691,  2453,  1073,  325,   73,  10,   1,  0, 0;

11: 56,  4815, 13004, 10357,  5058, 1836,  467,  91,  11,  1, 0, 0;

12: 77, 12867, 46452, 43462, 23953, 9631, 2941, 645, 111, 12, 1, 0, 0;

...

MAPLE

b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(add(

      `if`(i=1 or l[i-1]>l[i], (p->`if`(i<v, add(coeff(p, x, h)*

      `if`(h<v-i, x^(v-i), x^h), h=0..max(v-i, degree(p))), p))

       (b(n-1, i, subsop(i=l[i]+1, l))), 0), i=1..nops(l))+

       b(n-1, nops(l)+1, [l[], 1])))

    end:

T:= n-> (p->seq(coeff(p, x, i), i=0..n))(b(n-1, 1, [1])):

seq(T(n), n=0..12);

MATHEMATICA

b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Expand[Sum[If[i == 1 || l[[i-1]] > l[[i]], Function[{p}, If[i<v, Sum[Coefficient[p, x, h]* If[h < v-i, x^(v-i), x^h], {h, 0, Max[v-i, Exponent[p, x]]}], p]][b[n-1, i, ReplacePart[l, i -> l[[i]]+1]]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, 1, {1}]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Jan 07 2015, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000041, A244197, A244198, A244199, A244200, A244201, A244202, A244203, A244204, A244205, A244206.

Row sums are A000085.

Cf. A238129.

Sequence in context: A180969 A259479 A238343 * A238121 A171380 A170980

Adjacent sequences:  A238125 A238126 A238127 * A238129 A238130 A238131

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 21 2014

STATUS

approved

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Last modified April 27 04:48 EDT 2017. Contains 285507 sequences.