×

Theory and application of plane partitions. I. (English) Zbl 0225.05011


MSC:

05A17 Combinatorial aspects of partitions of integers
11P81 Elementary theory of partitions
20B30 Symmetric groups

Citations:

Zbl 0225.05012
Full Text: DOI

Online Encyclopedia of Integer Sequences:

Number of planar partitions of n decreasing across rows.

References:

[1] Ayoub, An Introduction to the Analytic Theory of Numbers (1963) · Zbl 0128.04303
[2] Birkhoff, Lattice Theory, 3. ed. (1967)
[3] Curtis, Representation Theory of Finite Groups and Associative Algebras (1962) · Zbl 0131.25601
[4] Feller, An Introduction to Probability Theory and its Applications I, 2. ed. (1957) · Zbl 0077.12201
[5] Hall, The Theory of Groups (1959) · Zbl 0084.02202
[6] Hardy, An Introduction to the Theory of Numbers, 4. ed. (1960) · Zbl 0086.25803
[7] Knuth, The Art of Computer Programming I (1968) · Zbl 0191.17903
[8] Littlewood, The Theory of Group Characters, 2. ed. (1950) · Zbl 0038.16504
[9] MacMahon, Combinatory Analysis 1 (1915)
[10] Muir, Theory of Determinants 1 (1906)
[11] Riordan, An Introduction to Combinatorial Analysis (1958) · Zbl 0078.00805
[12] Robinson, Representation Theory of the Symmetric Group (1961) · Zbl 0102.02002
[13] Rutherford, Substitutional Analysis (1948)
[14] Takács, Combinatorial Methods in the Theory of Stochastic Processes (1967)
[15] Aitken, On determinants of symmetric functions, Proc. Edin. Math. Soc. 1 (2) pp 55– (1929)
[16] Aitken, Note on dual symmetric functions, Proc. Edin. Math. Soc. 2 (2) pp 164– (1931) · Zbl 0001.11408 · doi:10.1017/S0013091500007719
[17] Atkin, Some computations for m-dimensional partitions, Proc. Cambridge Phil. Soc. 63 pp 1097– (1967) · Zbl 0157.03103 · doi:10.1017/S0305004100042171
[19] Bertrand, Solution d’un problème, C. R. Acad. Sci. Paris 105 pp 369– (1887)
[20] Carlitz, Some determinants of q-binomial coefficients, J. Reine Angew. Math. 226 pp 216– (1967) · Zbl 0162.03004
[21] Carlitz, Rectangular arrays and plane partitions, Acta Arith. 13 pp 29– (1967) · Zbl 0168.01502
[22] Carlitz, Enumeration of certain two-line arrays, Duke Math. Journ. 32 pp 529– (1966) · Zbl 0132.25103 · doi:10.1215/S0012-7094-65-03254-0
[23] Chaundy, Partition-generating functions, Quart. Journ. Math. (Oxford) 2 pp 234– (1931) · Zbl 0002.38203 · doi:10.1093/qmath/os-2.1.234
[24] Cheema, Some remarks on two- and three-line partitions, Duke Math. Journ. 31 pp 267– (1964) · Zbl 0117.28204 · doi:10.1215/S0012-7094-64-03125-4
[25] Chowla, On recursions connected with symmetric groups I, Canadian J. Math. 3 pp 328– (1951) · Zbl 0043.25904 · doi:10.4153/CJM-1951-038-3
[26] Erdös, A combinatorial problem in geometry, Composito Mathematica 2 pp 463– (1935)
[27] Farahat, On Schur functions, Proc. London Math. Soc. 8 (3) pp 621– (1956) · Zbl 0073.25705
[28] Frame, The hook lengths of, Canadian J. Math. 6 pp 316– (1954) · Zbl 0055.25404 · doi:10.4153/CJM-1954-030-1
[29] Frobenius, Sitzungsberichte Königl pp 516– (1900)
[30] Gordon, Two new representations of the partition function, Proc. Amer. Math. Soc. 13 pp 869– (1962) · Zbl 0107.04002 · doi:10.1090/S0002-9939-1962-0141607-5
[31] Gordon, Notes on Plane Partitions I, II, J. Combinatorial Theory 4 pp 72– (1968) · Zbl 0153.32803 · doi:10.1016/S0021-9800(68)80088-2
[32] Gordon, Notes on Plane Partitions III, Duke Math. Journ. 36 pp 801– (1969) · Zbl 0194.32302 · doi:10.1215/S0012-7094-69-03696-5
[34] Green, The characters of the finite general linear group, Trans. Amer. Math. Soc. 80 (1955) · Zbl 0068.25605 · doi:10.1090/S0002-9947-1955-0072878-2
[35] Hall, Proc. 4th Canad. Math. Congr. Banff. 1957 pp 147– (1959)
[36] Hardy, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17 (1918) · JFM 46.0198.02
[37] Jacobi, De functionibus alternantibus, Crelle’s Journal 22 pp 360– (1841) · ERAM 022.0687cj · doi:10.1515/crll.1841.22.360
[38] Klein, The Hall polynomial, J. Algebra 12 pp 61– (1969) · Zbl 0177.04103 · doi:10.1016/0021-8693(69)90017-9
[39] Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 pp 709– (1970) · Zbl 0199.31901 · doi:10.2140/pjm.1970.34.709
[40] Kosta, Ueber die Bestimmung von symmetrischen Functionen, Crelle’s Journal 81 (1876)
[41] Kosta, Über symmetrische Functionen, Crelle’s Journal 93 pp 89– (1882)
[42] Kosta, Tafeln und Formeln für symmetrische Funktionen, Jahresb. d. deutschen Math.-Verein. 16 pp 429– (1907)
[43] Kreweras, Sur une classe de problèmes de dénombrement lies au trellis des partitions des entiers, Cahiers du BURO 6 (1965)
[44] Littlewood, II. Group characters and algebra, Philosophical Trans. R. S. A-233 pp 99– (1934) · Zbl 0009.20203 · doi:10.1098/rsta.1934.0015
[45] MacMahon, Memoir on the theory of the partitions of numbers-Part IV, Philosophical Trans. R. S. A-209 pp 153– (1909) · JFM 40.0281.04 · doi:10.1098/rsta.1909.0007
[46] Moser, On solutions of xd = 1 in symmetric groups, Canadian J. Math. 7 pp 159– (1955) · Zbl 0064.02601 · doi:10.4153/CJM-1955-021-8
[47] Naegelbasch, Ueber eine Classe symmetrischen Functionen, Sch.-Prog. (1871)
[48] Nakayama, On some modular properties of irreducible representations of , I, II, Jap. J. Maths. 17 pp 89– (1940) · Zbl 0061.04001
[49] Nanda, Partition theory and thermodynamics of multi-dimensional oscillator assemblies, Proc. Cambridge Philos. Soc. 47 pp 591– (1951) · Zbl 0043.19605 · doi:10.1017/S0305004100026980
[50] Nanda, Tables of solid partitions, Proc. Nat. Inst. Sci. India 19 pp 313– (1953)
[51] Rademacher, On the partition function, Proc. London Math. Soc. 43 pp 241– (1937) · Zbl 0017.05503
[52] Read, The use of S-functions in combinatorial analysis, Canadian J. Math. 20 pp 808– (1968) · Zbl 0165.32802 · doi:10.4153/CJM-1968-080-x
[53] Robinson, On the representations of the symmetric group, Amer. J. Math. 60 pp 745– (1938) · JFM 64.0070.01 · doi:10.2307/2371609
[54] Rogers, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc. 16 pp 315– (1917) · JFM 46.0109.01
[55] Schensted, Longest increasing and decreasing subsequences, Canadian J. Math. 13 pp 179– (1961) · Zbl 0097.25202 · doi:10.4153/CJM-1961-015-3
[56] Schur, Inaugural-Dissertation (1901)
[57] Schützenberger, Quelques remarques sur une construction de Schensted, Math. Scand. 12 pp 117– (1963) · Zbl 0216.30202 · doi:10.7146/math.scand.a-10676
[58] Stanley, Solution to problem 5641 (proposed by Stanley Rabinowitz), Amer. Math. Monthly 76 pp 1153– (1969)
[61] Sudler, A direct proof of two theorems on two line partitions, Proc. Amer. Math. Soc. 16 pp 161– (1965) · Zbl 0127.27401 · doi:10.1090/S0002-9939-1965-0175798-X
[62] Trudi, Intorno un determinante pui generale, Rendic. dell’ Accad. (1864)
[63] Wright, Asymptotic partition formulae. I. Plane partitions, Quart, J. of Math. 2 pp 177– (1931) · Zbl 0002.38202 · doi:10.1093/qmath/os-2.1.177
[64] Wright, The generating function of solid partitions, Proc. Royal Soc. Edin. A-67 pp 185– (1966) · Zbl 0145.01502
[65] Wright, Rotatable partitions, J. London Math. Soc. 43 pp 501– (1968) · Zbl 0155.37101 · doi:10.1112/jlms/s1-43.1.501
[66] Young, Quantitative substitutional analysis I-IX, Proc. London Math. Soc. 33 (1) pp 97– (1901) · JFM 32.0157.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.