A262132
Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value <= 7.
Original entry on oeis.org
1, 1, 1, 3, 9, 45, 225, 1575, 11025, 99224, 893015, 9821203, 108031161, 1403167368, 18239614827, 273001402026, 4094100409041, 69325319221265, 1178005630046455, 22247669184972959, 422391796578333725, 8797758481967862297, 184548615259788777783
Offset: 0
A262133
Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value <= 8.
Original entry on oeis.org
1, 1, 1, 3, 9, 45, 225, 1575, 11025, 99225, 893024, 9823264, 108051897, 1404674661, 18256733309, 273850999635, 4104997257175, 69784953371975, 1184628370304553, 22507939035786507, 426574822311932257, 8958071268550577397, 187401362296416433302
Offset: 0
A262134
Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value <= 9.
Original entry on oeis.org
1, 1, 1, 3, 9, 45, 225, 1575, 11025, 99225, 893025, 9823274, 108056013, 1404720092, 18261352963, 273907724376, 4108601566641, 69834315000567, 1187167554049581, 22546611220735671, 428370636959431545, 8988329939437212520, 188741185272961560031
Offset: 0
A321280
Number T(n,k) of permutations p of [n] with exactly k descents such that the up-down signature of p has nonnegative partial sums; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/2)), read by rows.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 8, 1, 22, 22, 1, 52, 172, 1, 114, 856, 604, 1, 240, 3488, 7296, 1, 494, 12746, 54746, 31238, 1, 1004, 43628, 330068, 518324, 1, 2026, 143244, 1756878, 5300418, 2620708, 1, 4072, 457536, 8641800, 43235304, 55717312, 1, 8166, 1434318, 40298572, 309074508, 728888188, 325024572
Offset: 0
Triangle T(n,k) begins:
1;
1;
1;
1, 2;
1, 8;
1, 22, 22;
1, 52, 172;
1, 114, 856, 604;
1, 240, 3488, 7296;
1, 494, 12746, 54746, 31238;
1, 1004, 43628, 330068, 518324;
1, 2026, 143244, 1756878, 5300418, 2620708;
1, 4072, 457536, 8641800, 43235304, 55717312;
1, 8166, 1434318, 40298572, 309074508, 728888188, 325024572;
1, 16356, 4438540, 180969752, 2026885824, 7589067592, 8460090160;
...
- Alois P. Heinz, Rows n = 0..100, flattened
- S. Spiro, Ballot Permutations, Odd Order Permutations, and a New Permutation Statistic, arXiv preprint arXiv:1810.00993 [math.CO], 2018.
- David G. L. Wang, T. Zhao, The peak and descent statistics over ballot permutations, arXiv:2009.05973 [math.CO], 2020.
-
b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, 1/x,
add(expand(x*b(u-j, o-1+j, c-1)), j=1..u)+
add(b(u+j-1, o-j, c+1), j=1..o)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(`if`(n=0, 1, b(n, 0, 1))):
seq(T(n), n=0..14);
-
b[u_, o_, c_] := b[u, o, c] = If[c < 0, 0, If[u + o == 0, 1/x, Sum[Expand[ x*b[u - j, o - 1 + j, c - 1]], {j, 1, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ If[n == 0, 1, b[n, 0, 1]]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 08 2018, after Alois P. Heinz *)
A262135
Number of permutations p of [n] such that the up-down signature of p has nonnegative partial sums with a maximal value <= 10.
Original entry on oeis.org
1, 1, 1, 3, 9, 45, 225, 1575, 11025, 99225, 893025, 9823275, 108056024, 1404728312, 18261451790, 273921776850, 4108787293579, 69849383990843, 1187387325581683, 22560359186051977, 428591632052426778, 9000424273100962338, 188954690005716285164
Offset: 0