A279294
Number of permutations of [n] having exactly five (possibly overlapping) doubledescents.
Original entry on oeis.org
1, 14, 597, 11486, 265354, 5307274, 110325975, 2250025170, 46909998946, 990648740298, 21419845540563, 473829924582378, 10757656204264520, 250739074393237162, 6005432933786523655, 147819656394632180730, 3739797338380838531829, 97237232835306526768108
Offset: 7
A279295
Number of permutations of [n] having exactly six (possibly overlapping) doubledescents.
Original entry on oeis.org
1, 16, 1138, 25999, 775195, 18523314, 463088628, 11068300437, 268537592975, 6514804501312, 160541447975710, 4013204668037233, 102267326625073183, 2658365720541599806, 70614284743773530148, 1917874812704472161265, 53294689741358893762638
Offset: 8
A279296
Number of permutations of [n] having exactly seven (possibly overlapping) doubledescents.
Original entry on oeis.org
1, 18, 2195, 57980, 2234121, 62821766, 1877987735, 52129260840, 1462714640951, 40503784352590, 1131481211125461, 31802909031355052, 905596491625548929, 26146644212681105382, 767446589270204991395, 22921754812832515642328, 697455004277912572544814
Offset: 9
A279297
Number of permutations of [n] having exactly eight (possibly overlapping) doubledescents.
Original entry on oeis.org
1, 20, 4280, 127917, 6402984, 208910728, 7440888001, 237920318012, 7682498246390, 241437687882797, 7610900307811180, 239466123419848086, 7589962647890648579, 242520044163828141976, 7839783867491553405462, 256717246780820798056943, 8529320594216231644881865
Offset: 10
A279298
Number of permutations of [n] having exactly nine (possibly overlapping) doubledescents.
Original entry on oeis.org
1, 22, 8417, 279850, 18342412, 685212762, 29029313611, 1061113063498, 39275632545209, 1393699927780248, 49383101195522106, 1732429335364441968, 60909095747242747307, 2146854079671345945398, 76207802610167861527865, 2728392333397142478341958
Offset: 11
A279299
Number of permutations of [n] having exactly ten (possibly overlapping) doubledescents.
Original entry on oeis.org
1, 24, 16654, 607995, 52687209, 2225347678, 112136485958, 4652156989635, 196794669764694, 7848158942463262, 311532163358831142, 12142236428089109402, 472096448836605015907, 18301483090810860748890, 711486425957970444213722, 27780774432495984165279729
Offset: 12
A334257
Triangle read by rows: T(n,k) is the number of ordered pairs of n-permutations with exactly k common double descents, n>=0, 0<=k<=max{0,n-2}.
Original entry on oeis.org
1, 1, 4, 35, 1, 545, 30, 1, 13250, 1101, 48, 1, 463899, 51474, 2956, 70, 1, 22106253, 3070434, 217271, 7545, 96, 1, 1375915620, 229528818, 19372881, 864632, 20322, 126, 1, 108386009099, 21107789247, 2070917370, 113587335, 3530099, 61089, 160, 1
Offset: 0
T(4,1) = 30: There are 9 such ordered pairs formed from the permutations 3421,2431,1432. There are 9 such ordered pairs formed from the permutations 4312,4213,3214. Then pairing each of these 6 permutations with 4321 gives 12 more ordered pairs with exactly 1 common double descent. 9+9+12 = 30.
Triangle T(n,k) begins:
1;
1;
4;
35, 1;
545, 30, 1;
13250, 1101, 48, 1;
463899, 51474, 2956, 70, 1;
...
- R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, example 3.18.3e, page 366.
-
b:= proc(n, u, v, t) option remember; expand(`if`(n=0, 1,
add(add(b(n-1, u-j, v-i, x)*t, i=1..v)+
add(b(n-1, u-j, v+i-1, 1), i=1..n-v), j=1..u)+
add(add(b(n-1, u+j-1, v-i, 1), i=1..v)+
add(b(n-1, u+j-1, v+i-1, 1), i=1..n-v), j=1..n-u)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2, 1)):
seq(T(n), n=0..10); # Alois P. Heinz, Apr 26 2020
-
nn = 8; a = Apply[Plus,Table[Normal[Series[y x^3/(1 - y x - y x^2), {x, 0, nn}]][[n]]/(n +2)!^2, {n, 1, nn - 2}]] /. y -> y - 1; Map[Select[#, # > 0 &] &,
Range[0, nn]!^2 CoefficientList[Series[1/(1 - x - a), {x, 0, nn}], {x, y}]] // Grid
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