Restricted partitions and q-Pell numbers

Toufik Mansour; Mark Shattuck

Open Mathematics (2011)

  • Volume: 9, Issue: 2, page 346-355
  • ISSN: 2391-5455

Abstract

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In this paper, we provide new combinatorial interpretations for the Pell numbers p n in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by p n. By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of p n. Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.

How to cite

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Toufik Mansour, and Mark Shattuck. "Restricted partitions and q-Pell numbers." Open Mathematics 9.2 (2011): 346-355. <http://eudml.org/doc/268972>.

@article{ToufikMansour2011,
abstract = {In this paper, we provide new combinatorial interpretations for the Pell numbers p n in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by p n. By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of p n. Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.},
author = {Toufik Mansour, Mark Shattuck},
journal = {Open Mathematics},
keywords = {Pattern avoidance; Inversion; Comajor index; Pell number; q-generalization; pattern avoidance; inversion; comajor index; -generalization},
language = {eng},
number = {2},
pages = {346-355},
title = {Restricted partitions and q-Pell numbers},
url = {http://eudml.org/doc/268972},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Toufik Mansour
AU - Mark Shattuck
TI - Restricted partitions and q-Pell numbers
JO - Open Mathematics
PY - 2011
VL - 9
IS - 2
SP - 346
EP - 355
AB - In this paper, we provide new combinatorial interpretations for the Pell numbers p n in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by p n. By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of p n. Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.
LA - eng
KW - Pattern avoidance; Inversion; Comajor index; Pell number; q-generalization; pattern avoidance; inversion; comajor index; -generalization
UR - http://eudml.org/doc/268972
ER -

References

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  2. [2] Benjamin AT., Plott S.P., Sellers J.A., Tiling proofs of recent sum identities involving Pell numbers, Ann. Comb., 2008, 12(3), 271–278 http://dx.doi.org/10.1007/s00026-008-0350-5 Zbl1169.05305
  3. [3] Briggs K.S., Little D.P., Sellers J.A., Tiling proofs of various g-Pell identities via tilings, Ann. Comb. (in press) Zbl1233.05037
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  8. [8] Knuth D.E., The Art of Computer Programming, Vol. 1,3, Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley, Reading, 1968, 1974 Zbl0191.17903
  9. [9] Sagan B.E., Pattern avoidance in set partitions, Ars Combin., 2010, 94(1), 79–96 Zbl1240.05018
  10. [10] Santos J.P.O., Sills A.V., g-Pell sequences and two identities of V. A. Lebesgue, Discrete Math., 2002, 257(1), 125–142 http://dx.doi.org/10.1016/S0012-365X(01)00475-7 Zbl1007.05017
  11. [11] Simion R., Schmidt F.W., Restricted permutations, European J. Combin., 1985, 6(4), 383–406 Zbl0615.05002
  12. [12] Sloane N.J.A., The On-Line Encyclopedia of Integer Sequences, http://oeis.org Zbl1274.11001
  13. [13] Stanley R.P., Enumerative Combinatorics, Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole, Monterey, 1986 
  14. [14] Stanton D., White D., Constructive Combinatorics, Undergrad. Texts Math., Springer, New York, 1986 

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