A316390
Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of three.
Original entry on oeis.org
1, 3, 28, 130, 1263, 8090, 88101, 724189, 8887448, 89401804, 1229179691, 14638611036, 223711095367, 3078744103979, 51892788554614, 810254535452378, 14955918856848519, 261173044555806630, 5245841953983851853, 101285541723126490941, 2201267668629421856324
Offset: 3
-
b:= proc(u, o, c, k) option remember;
`if`(c<0 or c>k, 0, `if`(u+o=0, 1,
add(b(u-j, o-1+j, c+1, k), j=1..u)+
add(b(u+j-1, o-j, c-1, k), j=1..o)))
end:
a:= n-> b(n, 0$2, 3)-b(n, 0$2, 2):
seq(a(n), n=3..23);
A316391
Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of four.
Original entry on oeis.org
1, 4, 62, 340, 4734, 33855, 495371, 4403025, 70083872, 746704117, 13023762276, 161905131484, 3091115525637, 43928623624790, 914530883776894, 14623431780216366, 330413968185491070, 5870376151413374683, 143271256595612492851, 2799645366893284489691
Offset: 4
-
b:= proc(u, o, c, k) option remember;
`if`(c<0 or c>k, 0, `if`(u+o=0, 1,
add(b(u-j, o-1+j, c+1, k), j=1..u)+
add(b(u+j-1, o-j, c-1, k), j=1..o)))
end:
a:= n-> b(n, 0$2, 4)-b(n, 0$2, 3):
seq(a(n), n=4..23);
A316392
Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of five.
Original entry on oeis.org
1, 5, 129, 819, 16066, 127538, 2423226, 23367449, 459383574, 5246611332, 109138956326, 1446115120862, 32069014233249, 484780196858918, 11478459399841878, 195255855453716821, 4931560739013573590, 93326559046408832001, 2509294817575539112099
Offset: 5
-
b:= proc(u, o, c, k) option remember;
`if`(c<0 or c>k, 0, `if`(u+o=0, 1,
add(b(u-j, o-1+j, c+1, k), j=1..u)+
add(b(u+j-1, o-j, c-1, k), j=1..o)))
end:
a:= n-> b(n, 0$2, 5)-b(n, 0$2, 4):
seq(a(n), n=5..23);
A316393
Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of six.
Original entry on oeis.org
1, 6, 261, 1890, 52022, 455231, 11174035, 116105272, 2810232512, 34036483163, 844691910962, 11731978216291, 303637667232802, 4769379288424677, 129700918311614279, 2277005590881369266, 65261900211279910831, 1267764017301809851710, 38324737795523150842616
Offset: 6
-
b:= proc(u, o, c, k) option remember;
`if`(c<0 or c>k, 0, `if`(u+o=0, 1,
add(b(u-j, o-1+j, c+1, k), j=1..u)+
add(b(u+j-1, o-j, c-1, k), j=1..o)))
end:
a:= n-> b(n, 0$2, 6)-b(n, 0$2, 5):
seq(a(n), n=6..24);
A316394
Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of seven.
Original entry on oeis.org
1, 7, 522, 4260, 163871, 1572713, 49601660, 554432537, 16431601190, 211104220038, 6214132488281, 90601727479330, 2718687446733807, 44477388811619142, 1378374571651666083, 25055072909382001747, 807272266530396465622, 16165637154045080226474
Offset: 7
-
b:= proc(u, o, c, k) option remember;
`if`(c<0 or c>k, 0, `if`(u+o=0, 1,
add(b(u-j, o-1+j, c+1, k), j=1..u)+
add(b(u+j-1, o-j, c-1, k), j=1..o)))
end:
a:= n-> b(n, 0$2, 7)-b(n, 0$2, 6):
seq(a(n), n=7..24);
A316395
Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of eight.
Original entry on oeis.org
1, 8, 1040, 9468, 507355, 5313447, 214378961, 2571977379, 92953037066, 1265907917962, 44038999833044, 674142774632948, 23379215615715958, 398561935596289153, 14037530250073013445, 264291741199540446059, 9551899031473405653870, 197148463934806397523934
Offset: 8
-
b:= proc(u, o, c, k) option remember;
`if`(c<0 or c>k, 0, `if`(u+o=0, 1,
add(b(u-j, o-1+j, c+1, k), j=1..u)+
add(b(u+j-1, o-j, c-1, k), j=1..o)))
end:
a:= n-> b(n, 0$2, 8)-b(n, 0$2, 7):
seq(a(n), n=8..25);
A316396
Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of nine.
Original entry on oeis.org
1, 9, 2071, 20845, 1553153, 17662969, 908651571, 11670560732, 512693233164, 7392808621010, 303061463720474, 4869546964922509, 194661534866479194, 3459210686800253224, 138131753631241199208, 2695708505172764233290, 109038227244360661170616
Offset: 9
-
b:= proc(u, o, c, k) option remember;
`if`(c<0 or c>k, 0, `if`(u+o=0, 1,
add(b(u-j, o-1+j, c+1, k), j=1..u)+
add(b(u+j-1, o-j, c-1, k), j=1..o)))
end:
a:= n-> b(n, 0$2, 9)-b(n, 0$2, 8):
seq(a(n), n=9..25);
A316397
Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of ten.
Original entry on oeis.org
1, 10, 4127, 45562, 4719041, 58003461, 3795919780, 52052335254, 2772611610514, 42268105493955, 2037044419366071, 34333238955892416, 1578674337291922196, 29239755067140936242, 1321207703588407017510, 26733500408009431631728, 1208079945873987947779946
Offset: 10
-
b:= proc(u, o, c, k) option remember;
`if`(c<0 or c>k, 0, `if`(u+o=0, 1,
add(b(u-j, o-1+j, c+1, k), j=1..u)+
add(b(u+j-1, o-j, c-1, k), j=1..o)))
end:
a:= n-> b(n, 0$2, 10)-b(n, 0$2, 9):
seq(a(n), n=10..26);