A263776
Triangle read by rows: T(n,k) (n>=0, 0<=k<=A002620(n-1)) is the number of permutations of [n] with k nestings.
Original entry on oeis.org
1, 1, 2, 5, 1, 14, 8, 2, 42, 45, 25, 7, 1, 132, 220, 198, 112, 44, 12, 2, 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1, 1430, 4368, 7280, 8400, 7460, 5392, 3262, 1664, 716, 256, 74, 16, 2, 4862, 18564, 38556, 56100, 63648, 59670, 47802, 33338, 20466, 11115
Offset: 0
Triangle begins:
0 : 1;
1 : 1;
2 : 2;
3 : 5, 1;
4 : 14, 8, 2;
5 : 42, 45, 25, 7, 1;
6 : 132, 220, 198, 112, 44, 12, 2;
7 : 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1;
...
- Alois P. Heinz, Rows n = 0..50, flattened
- A. Claesson and T. Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001.
- S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math 38 (2007) 149-163.
- FindStat - Combinatorial Statistic Finder, The number of nestings of a permutation, The number of crossings of a permutation
- R. Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
- Lucas Sá and Antonio M. García-García, The Wishart-Sachdev-Ye-Kitaev model: Q-Laguerre spectral density and quantum chaos, arXiv:2104.07647 [hep-th], 2021.
Columns k=0-10 give:
A000108,
A002696,
A094218,
A094219,
A120812,
A120813,
A120814,
A120815,
A120816,
A264496,
A264497.
-
b:= proc(u, o) option remember;
`if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
add(expand(b(u+j-1, o-j)*x^(j-1)), j=1..o))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10); # Alois P. Heinz, Nov 14 2015
-
b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Expand[b[u+j-1, o-j]*x^(j-1)], {j, 1, o}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[ T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)
A260665
Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 12-3; triangle T(n,k), n>=0, 0<=k<=(n-1)*(n-2)/2-[n=0], read by rows.
Original entry on oeis.org
1, 1, 2, 5, 1, 15, 7, 1, 1, 52, 39, 13, 12, 2, 1, 1, 203, 211, 112, 103, 41, 24, 17, 5, 2, 1, 1, 877, 1168, 843, 811, 492, 337, 238, 122, 68, 39, 28, 8, 5, 2, 1, 1, 4140, 6728, 6089, 6273, 4851, 3798, 2956, 1960, 1303, 859, 594, 314, 204, 110, 64, 43, 17, 8, 5, 2, 1, 1
Offset: 0
T(4,1) = 7: 1324, 1342, 2134, 2314, 2341, 3124, 4123.
T(4,2) = 1: 1243.
T(4,3) = 1: 1234.
T(5,3) = 12: 12534, 12543, 13245, 13425, 13452, 21345, 23145, 23415, 23451, 31245, 41235, 51234.
T(5,4) = 2: 12435, 12453.
T(5,5) = 1: 12354.
T(5,6) = 1: 12345.
Triangle T(n,k) begins:
0 : 1;
1 : 1;
2 : 2;
3 : 5, 1;
4 : 15, 7, 1, 1;
5 : 52, 39, 13, 12, 2, 1, 1;
6 : 203, 211, 112, 103, 41, 24, 17, 5, 2, 1, 1;
7 : 877, 1168, 843, 811, 492, 337, 238, 122, 68, 39, 28, 8, 5, 2, 1, 1;
Columns k=0-10 give:
A000110,
A092923,
A264451,
A264452,
A264453,
A264454,
A264455,
A264456,
A264457,
A264458,
A264459.
-
b:= proc(u, o) option remember;
`if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
add(expand(b(u+j-1, o-j)*x^(o-j)), j=1..o))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);
-
b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1], {j, 1, u}] + Sum[Expand[b[u + j - 1, o - j]*x^(o - j)], {j, 1, o}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0] ]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)
A264460
Number of permutations of [n] with exactly one occurrence of the generalized pattern 23-1.
Original entry on oeis.org
1, 6, 32, 171, 944, 5444, 32919, 208816, 1388240, 9657929, 70187054, 531857288, 4194927585, 34379859346, 292303350268, 2574284790795, 23450837821836, 220681535036288, 2142618638738279, 21438586249394500, 220827871704427308, 2339281577294955745
Offset: 3
a(3) = 1: 231.
a(4) = 6: 1342, 2314, 2413, 2431, 3241, 4231.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^u, x, 2), polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 1):
seq(a(n), n=3..25);
-
b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Series[b[u+j-1, o-j] x^u, {x, 0, 2}] // Normal, {j, 1, o}]];
a[n_] := Coefficient[b[n, 0], x, 1];
a /@ Range[3, 25] (* Jean-François Alcover, Sep 28 2020, after Maple *)
A264461
Number of permutations of [n] with exactly two (possibly overlapping) occurrences of the generalized pattern 23-1.
Original entry on oeis.org
3, 23, 152, 984, 6460, 43626, 304939, 2211467, 16649780, 130097338, 1054226016, 8850736900, 76901730751, 690749091147, 6406953787268, 61300205459232, 604367205789092, 6133919028981542, 64027105979768111, 686736004045762143, 7562191796264603160
Offset: 4
a(4) = 3: 2341, 3412, 3421.
a(5) = 23: 13452, 14523, 14532, 23415, 23514, 23541, 24351, 25341, 32451, 34125, 34152, 34215, 35124, 35142, 35214, 35412, 35421, 42351, 43512, 43521, 52341, 53412, 53421.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^u, x, 3), polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 2):
seq(a(n), n=4..25);
-
b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Series[b[u+j-1, o-j] x^u, {x, 0, 3}] // Normal, {j, 1, o}]];
a[n_] := Coefficient[b[n, 0], x, 2];
a /@ Range[4, 25] (* Jean-François Alcover, Sep 28 2020, after Maple *)
A264462
Number of permutations of [n] with exactly three (possibly overlapping) occurrences of the generalized pattern 23-1.
Original entry on oeis.org
10, 98, 791, 6082, 46508, 360376, 2855580, 23244370, 194796665, 1682243510, 14973751272, 137345968648, 1297620307926, 12620234271634, 126265094967383, 1298634572382618, 13720327052477584, 148800093788609512, 1655374072150829472, 18877499531817214482
Offset: 5
a(5) = 10: 23451, 24513, 24531, 34251, 35241, 45123, 45132, 45213, 45312, 45321.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^u, x, 4), polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 3):
seq(a(n), n=5..25);
A264463
Number of permutations of [n] with exactly four (possibly overlapping) occurrences of the generalized pattern 23-1.
Original entry on oeis.org
3, 62, 624, 5513, 46880, 396545, 3388893, 29497132, 262619041, 2397130075, 22458202716, 216063700398, 2134707078315, 21655074753266, 225471830153938, 2408463527492433, 26380434283682048, 296130434080550183, 3404883834560222589, 40077202708717399396
Offset: 5
a(5) = 3: 34512, 34521, 45231.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^u, x, 5), polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 4):
seq(a(n), n=5..25);
A264464
Number of permutations of [n] with exactly five (possibly overlapping) occurrences of the generalized pattern 23-1.
Original entry on oeis.org
22, 392, 4302, 41979, 393476, 3652779, 34099448, 322746351, 3111556310, 30637025849, 308542096284, 3180677263083, 33574281993047, 362907367462521, 4016295885663247, 45496983534864142, 527373693673894161, 6252643277063017082, 75794122449729562363
Offset: 6
a(6) = 22: 245613, 245631, 256341, 345261, 346251, 356124, 356142, 356214, 356412, 356421, 452361, 453612, 453621, 462351, 463512, 463521, 561342, 562314, 562413, 562431, 563241, 564231.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^u, x, 6), polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 5):
seq(a(n), n=6..25);
A264465
Number of permutations of [n] with exactly six (possibly overlapping) occurrences of the generalized pattern 23-1.
Original entry on oeis.org
11, 240, 3328, 36774, 377610, 3775281, 37545239, 375714151, 3808210872, 39250117725, 412320359485, 4420852570619, 48416817503118, 541846406216437, 6197311080340307, 72436698579083259, 865110836456018947, 10554489471399030408, 131500190867301682844
Offset: 6
a(6) = 11: 345612, 345621, 356241, 456123, 456132, 456213, 456312, 456321, 562341, 563412, 563421.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^u, x, 7), polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 6):
seq(a(n), n=6..25);
A264466
Number of permutations of [n] with exactly seven (possibly overlapping) occurrences of the generalized pattern 23-1.
Original entry on oeis.org
1, 111, 2141, 29066, 336137, 3663730, 39037155, 414003430, 4414401391, 47606401475, 521147444815, 5804080039932, 65854473968668, 761861585833395, 8990938684395442, 108258685340444377, 1330041483208589743, 16671576238569453924, 213169773223813966680
Offset: 6
a(6) = 1: 456231.
a(7) = 111: 1567342, 2456713, 2456731, ..., 6753412, 6753421, 7456231.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^u, x, 8), polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 7):
seq(a(n), n=6..25);
A264467
Number of permutations of [n] with exactly eight (possibly overlapping) occurrences of the generalized pattern 23-1.
Original entry on oeis.org
55, 1393, 22180, 290341, 3443222, 39212955, 439762906, 4922646194, 55440198721, 631281758915, 7290146619344, 85550258610256, 1021466597564828, 12418915430929303, 153814790549535203, 1941191617618161308, 24964754218212162461, 327159226788279647296
Offset: 7
a(7) = 55: 2567341, 3456712, 3456721, ..., 6745213, 6745312, 6745321.
-
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(
b(u-j, o+j-1), j=1..u) +add(convert(series(
b(u+j-1, o-j)*x^u, x, 9), polynom), j=1..o))
end:
a:= n-> coeff(b(n, 0), x, 8):
seq(a(n), n=7..25);
Showing 1-10 of 12 results.
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